Sequencing and Counting with the multicost-regular Constraint

Transcription

1 Sequencing and Counting with the multicot-regular Contraint Julien Menana and Sophie Demaey École de Mine de Nante, LINA CNRS UMR 6241, F Nante, France. Abtract. Thi paper introduce a global contraint encapulating a regular contraint together with everal cumulative cot. It i motivated in the context of peronnel cheduling problem, where a chedule meet pattern and occurrence requirement which are intricately bound. The optimization problem underlying the multicot-regular contraint i NP-hard but it admit an efficient Lagrangian relaxation. Hence, we propoe a filtering baed on thi relaxation. The expreivene and the efficiency of thi new contraint i experimented on peronnel cheduling benchmark intance with tandard work regulation. The comparative empirical reult how how multicot-regular can ignificantly outperform a decompoed model with regular and global-cardinality contraint. 1 Introduction Many combinatorial deciion problem involve the imultaneou action of equencing and counting object, epecially in the large cla of routing and cheduling problem. In routing, a vehicle viit a equence of location following a path in the road network according to ome numerical requirement on the whole travelling ditance, the time pent, or the vehicle capacity. If only one numerical attribute i pecified, finding a route i to olve a hortet/longet path problem. For everal attribute, the problem a Reource Contrained Shortet/Longet Path Problem (RCSPP) become NP-hard. All thee numerical requirement may dratically retrict the et of path in the network which correpond to the actual valid route. Hence, it i much more efficient to take thee requirement into account throughout the earch of a path, rather than each eparately. Peronnel cheduling problem can be treated analogouly. Planning a worker chedule i to equence activitie (or hift) over a time horizon according to many variou work regulation, a for example: a working night i followed by a free morning, a night hift cot twice a much a a day hift, at leat 10 day off a month, etc. Hence, a chedule meet both tructural requirement defined a allowed pattern of activitie and numerical requirement defined a aignment cot or counter which are intricately bound. Modelling thee requirement individually i itelf a hard tak, for which the expreivene and the flexibility of Contraint Programming (CP) i recognized. Modelling thee requirement efficiently i till a harder tak a it mean to aggregate all

2 of them in order to proce thi et of tied requirement a a whole. By introducing the regular global contraint, Peant [1] ha propoed an elegant and efficient way to model and to enforce all the pattern requirement together. The allowed pattern are gathered in an acyclic digraph whoe path coincide with the valid equence of activitie. Thi approach wa later extended to optimization contraint oft-regular [2] and cot-regular [3] for enforcing bound on the global cot a violation cot or any financial cot of the equence of aignment. The underlying problem i now to compute hortet and longet path in the acyclic graph of pattern. The cot-regular contraint wa uccefully applied to olve real-world peronnel cheduling problem under a CP-baed column-generation approach [3]. Neverthele, the author complained about the weak interaction in their CP model between the cot-regular contraint and an external global-cardinality ued for modelling occurrence requirement. Actually, with uch a decompoition, the upport graph of cot-regular maintain many path which do not atify the cardinality contraint. In thi paper, we till generalize thi approach for handling everal cot attribute within one global contraint multicot-regular. Such a contraint allow to reaon imultaneouly on the equencing and counting requirement occurring in peronnel cheduling problem. A mentioned above, the underlying optimization problem i a RCSPP and it remain NP-hard even when the graph i acyclic. Hence, the filtering algorithm we preent achieve a relaxed level of conitency. It i baed on the Lagrangian relaxation of the RCSPP following the principle by Sellmann [4] for Lagrangian relaxation-baed filtering. Our implementation of multicot-regular i available in the ditribution of the open-ource CP olver CHOCO 1. The paper i organized a follow. In Section 2, we preent the cla of regular contraint and provide a theoretical comparion between the pathfinding approach of Peant [1] and the decompoition-baed approach of Beldiceanu et al. [5]. We introduce then the new contraint multicot-regular. In Section 3, we introduce the Lagrangian relaxation-baed filtering algorithm. In Section 4, we decribe a variety of tandard work regulation and invetigate a ytematic way of building one intance of multicot-regular from a et of requirement. In Section 5, comparative empirical reult on benchmark intance of peronnel cheduling problem are given. They how how multicot-regular can ignificantly outperform a decompoed model with regular and globalcardinality contraint. 2 Regular Language Memberhip Contraint In thi ection, we recall the definition of the regular contraint and report on related work, before introducing multicot-regular. Firt, we recall baic notion of automata theory and introduce notation ued throughout thi paper: We conider a non empty et Σ called the alphabet. Element of Σ are called ymbol, equence of ymbol are called word, and et of word are called 1

3 language over Σ. An automaton Π i a directed multigraph (Q, ) whoe arc are labelled by the ymbol of an alphabet Σ, and where two non-empty ubet of vertice I and A are ditinguihed. The et Q of vertice i called the et of tate of Π, I i the et of initial tate, and A i the et of accepting tate. The non-empty et Q Σ Q of arc i called the et of tranition of Π. A word in Σ i aid to be accepted by Π if it i the equence of the arc label of a path from an initial tate to an accepting tate in Π. Automaton Π i a determinitic finite automaton (DFA) if i finite and if it ha only one initial tate (I = {}) and no two tranition haring the ame initial extremity and the ame label. The language accepted by a FA i a regular language. 2.1 Path-Finding and Decompoition: Two Approache for regular The regular language memberhip contraint wa introduced by Peant in [1]. Given a equence X = (x 1, x 2,..., x n ) of finite domain variable and a determinitic finite automaton Π = (Q, Σ,, {}, A), the contraint regular(x, Π) hold iff X i a word of length n over Σ accepted by DFA Π. By definition, the olution of regular(x, Π) are in one-to-one correpondance with the path of exactly n arc connecting to a vertex in A in the directed multigraph Π. Let δ i denote the et of tranition that appear a the i-th arc of uch a path, then a value for x i i conitent iff δ i contain a tranition labelled by thi value. Coincidently, Peant [1] and Beldiceanu et al [5] introduce two orthogonal approache to achieve GAC on regular (ee Figure 1). The approach propoed by Peant [1] i to unfold Π a an acyclic DFA Π n which accept only the word of length n. By contruction, Π n i a layered multigraph with tate in layer 0 (the ource), the accepting tate A in layer n (the ink), and where the et of arc in any layer i coincide with δ i. A breadth-firt earch allow to maintain the coherence between Π n and the variable domain by pruning the arc in δ i whoe label are not in the domain of x i, then by pruning the vertice and arc which are not connected to a ource and to a ink. In Beldiceanu et al [5], a regular i decompoed a n tuple contraint for modelling the et δ 1, δ 2,..., δ n. The decompoition introduce tate variable q 0 {}, q 1,..., q n 1 Q, q n A and ue triplet relation defined in extenion to enforce GAC on the tranition contraint (q i 1, x i, q i ) δ i. Such a contraint network being Berge-acyclic, enforcing AC on the decompoition achieve GAC on regular. In the firt approach, a pecialized algorithm i defined to maintain all the upport path, while in the econd approach, the tranition are modeled with tuple contraint which are directly propagated by the CP olver. The two approache are orthogonal. Actually, the econd model may mimic the pecialized algorithm depending on the choen propagation. If we aume w.l.o.g. that Σ i the union of the variable domain, then the initial run of Peant algorithm for the contruction of Π n i performed in O(n ) time and pace (with Q Σ if Π i a DFA). Incremental filtering i performed with the ame wort-cae complexity with a forward/backward traveral of Π n. Actually, the complexity of the algorithm relie more on the ize n of the unfolded automaton Π n rather than on the ize of the pecified

4 a unfolded automaton: a a a b b b b b 1 a 1 a 1 1 b b a 1 decompoed model: (q 0, x 1, q 1) {(, a, ), (, b, 1)}, (q 1, x 2, q 2) {(, a, ), (1, a, 1)}, (q 2, x 3, q 3) {(, a, ), (1, b, ),////////////////////// (, b, 1), (1, a, 1)}, (q 3, x 4, q 4) {(, b, 1),////////// (1, b, )}. Fig. 1. Conider the DFA depicted above applied to X {a, b} {a} {a, b} {b}. The unfolded automaton of regular i depicted on the left and the decompoed model on the right. The dahed tranition are dicarded in both model. automaton Π. Note for intance that when the pecified automaton Π accept only word of length n then it i already unfolded (Π = Π n ) and the firt run of the algorithm i in O( ). In practice, a in our experiment (Section 5), Π n can even be much maller than Π, meaning that many accepting tate in Π cannot be reached in exactly n tranition. The incremental filtering i performed in O( n ) time with, in uch a cae, n n. regular i a very expreive contraint. It i ueful to model pattern contraint ariing in many planning problem, but alo to reformulate other global contraint [5] or to model tuple defined in extenion. An other application of regular i to model a liding contraint: recently, Beière et al. [6] have introduced the lide meta contraint. In it more general form, lide take a argument a matrix of variable Y of ize n p and a contraint C of arity pk with k n. lide(y, C) hold if and only of C(y 1 i+1,..., yp i+1,..., y1 i+k,..., yp i+k ) hold for 0 i n k. Uing the decompoition propoed in [5], regular(x, Π) can be reformulated a lide([q, X], C ), where Q i the equence of tate variable and C i the tranition contraint C (q, x, q, x ) (q, x, q ). Converely [6], a lide contraint can be reformulated a a regular but it may require to enumerate all valid tuple for C. Thi reformulation can however be ueful in the context of planning (epecially for car equencing) to model a liding cardinality contraint alo known a equence. Even if powerful pecialized algorithm exit for thi contraint (ee e.g. [7]), the automaton reulting from the reformulation can be integrated with other pattern requirement a we will how in Section 4. Finally, one hould notice the work (ee e.g. [8]) related to contextfree grammar contraint. Though, mot of the rule encountered in peronnel cheduling can be decribed uing regular language. 2.2 Maintaining Pattern with Cumulative Cot and Cardinalitie Peronnel cheduling problem are uually defined a optimization problem. Mot often, the criterion to optimize i a cumulative cot, i.e. the um of cot aociated to each aignment of a worker to a given activity at a given time.

5 Such a cot ha everal meaning: it can model a financial cot, a preference, or a value occurrence. Now, deigning a valid chedule for one worker i to enforce the equence of aignment to comply with a given pattern while enuring that the total cot of the aignment i bounded. Thi can be pecified by mean of a cot-regular contraint [3]. Given c = (c ia ) i [1..n] a Σ a matrix of real aignment cot and z [z, z] a bounded variable (z, z R), cot-regular(x, z, Π, c) hold iff regular(x, Π) hold and n i=1 c ix i = z. Note that it ha the knapack contraint [9] a a pecial cae and that, unle P = NP, one can enforce GAC on a knapack contraint at bet in peudo-polynomial time, i.e. the run time i polynomial in the value of the bound of z. A a conequence, enforcing GAC on cot-regular i NP-hard. The definition of cot-regular reveal a natural decompoition a a regular contraint channeled to a knapack contraint. Actually, it i equivalent to the decompoition propoed by Beldiceanu et al. [5] when dealing with one cumulative 2 cot: cot variable k i are now aociated to the previou tate variable q i, with k 0 = 0 and k n = z, and everal arithmetic and element contraint model the knapack and channeling contraint. In hort, thi formulation can be rewritten a lide([q, X, K], C c ), with Cc (q i 1, x i 1, k i 1, q i, x i, k i ) (q i 1, x i 1, q i ) k i = k i 1 +c ixi. Depending on the ize of the domain of the cot variable, GAC can be enforced on knapack in reaonable time. However, even in thi cae, ince the contraint hypergraph of the decompoed model i no longer Berge-acyclic but α-acyclic, one ha to enforce pairwie-conitency on the hared variable a pair (q i, k i ) of tate and cot variable of the tranition contraint in order to achieve GAC. A imilar option propoed for lide [6] i to enforce AC on the dual encoding of the hypergraph of the C c contraint, but again it require to explicit all the upport tuple and then, it may be of no practical ue. The filtering algorithm preented in [3] for cot-regular i a light adaptation 3 of Peant algorithm for regular. It i baed on the computation of hortet and longet path in the unfolded graph Π n valued by the tranition cot. To each vertex (i, q) in any layer i of Π n are aociated two bounded cot variable k iq and k+ iq modelling the length of the path repectively from layer 0 to (i, q) and from (i, q) to layer n. The cot variable can trivially be initialized during the contruction of Π n : k iq in the forward phae and k+ iq in the backward phae. The bound of variable z are then pruned according to the condition z k 0 +. Converely, an arc ((i 1, q), a, (i, q )) δ i can be removed whenever: k (i 1)q + c ia + k + iq > z or k (i 1)q + c ia + k + iq > z. A graph Π n i acyclic, maintaining the cot variable, i.e. hortet and longet path, can be performed by breadth-firt traveral with the ame time complexity O( n ) than for maintaining the connexity of the graph in regular. 2 The model in [5] can deal not only with um but alo with variou arithmetic function on cot, but no example of ue i provided. 3 Previouly, the algorithm wa partially for minimization only applied to the pecial cae oft-regular[hamming] in [5] and in [2].

6 A aid before, thi algorithm achieve a hybrid level of conitency on cot-regular. A a matter of fact, it enforce a ort of pairwie-conitency on the decompoed model between each tate variable and the bound of the aociated cot variable, according to the relation q i = (i, q) k i = k iq. Hence, it dominate the decompoed model knapack regular when only Bound Conitency i enforced on the cot variable. Otherwie, if AC i enforced on knapack then the two approache are incomparable a how the two example depicted in Figure 2 and 3. a [1] 1 a [1] 3 b [0] b [0] 2 x 1 {//a, b}, x 2 {//a, b}, z [0, 1]. [0,1] (q 0, x 1, q 1, j 1) {(, a, 1, 1), (, b, 2, 2)}, element(k 1, j 1, (k 0 + 1, k 0)), (q 1, x 2, q 2, j 2) {(1, a, 3, 1), (2, b, 3, 2)}, element(z, j 2, (k 1 + 1, k 1)), q 0 {}, q 1 {1, 2}, q 2 {3}, k 0 {0}, k 1 {0, 1}, x 1 {a, b}, x 2 {a, b}, z {0, 1}. Fig. 2. Conider the depicted DFA with cot in bracket applied to X = (x 1, x 2) {a, b} {a, b} and z [0, 1]. The cot-regular algorithm (on the left) dicard the dahed tranition and hence achieve GAC. The decompoed model (on the right) i arc-conitent but not globally conitent. a [0] a [0] c [1] 1 2 [2,2] b [2] b [2] x 1 {a, b, c}, x 2 {a, b}, z [2, 2]. (x 1, j 1) {(a, 1),/////// (c, 2), (b, 3)}, element(k 1, j 1, (k 0,///////// k 0 + 1, k 0 + 2)), (x 2, j 2) {(a, 1), (b, 2)}, element(z, j 2, (k 1, k 1 + 2)), k 0 {0}, k 1 {0,//1, 2}, x 1 {a, b,/c}, x 2 {a, b}, z {2}. Fig. 3. Conider now the depicted DFA applied to X = (x 1, x 2) {a, b, c} {a, b} and z [2, 2]. Enforcing AC on the decompoed model (on the right) achieve GAC. The cot-regular algorithm (on the left) doe not achieve GAC ince the minimum and maximum path travering arc x 1 = c are conitent with the bound on z. 2.3 The multicot-regular Contraint A natural generalization of cot-regular i to handle everal cumulative cot: given a vector Z = (z 0,..., z R ) of bounded variable and c = (c r ia )r [0..R] i [1..n],a Σ a matrix of aignment cot, multicot-regular(x, Z, Π, c) hold if and only if regular(x, Π) hold and n i=1 cr ix i = z r for all 0 r R. Such a generalization ha an important motivation in the context of peronnel cheduling. Actually, apart a financial cot and pattern retriction, an individual chedule

7 i uually ubject to a global-cardinality contraint bounding the number of occurrence of each value in the equence. Thee bound can dratically retrict the language on which the chedule i defined. Hence, it could be convenient to tackle them within the regular contraint in order to reduce the upport graph. A a generalization of cot-regular or of the global-equencing contraint [10], we cannot hope to achieve GAC in polynomial time here. Note that the model by Beldiceanu et al [5] and imilarly the lide contraint wa alo propoed for dealing with everal cot but again, it amount to decompoe a a regular contraint channeled with one knapack contraint for each cot. Hence, we ought to exploit the tructure of the upport graph of Π n to get a good relaxed propagation for multicot-regular. The optimization problem underlying cot-regular were hortet and longet path problem in Π n. The optimization problem underlying multicot-regular are now the Reource Contrained Shortet and Longet Path Problem (RCSPP and RCLPP) in Π n. The RCSPP (rep. RCLPP) i to find the hortet (rep. longet) path between a ource and a ink in a valued directed graph, uch that the quantitie of reource accumulated on the arc do not exceed ome limit. Even with one reource on acyclic digraph, thi problem i known to be NP-hard[11]. Two approache are mot often ued to olve RCSPP [11]: dynamic programming and Lagrangian relaxation. Dynamic programming-baed method extend the uual hort path algorithm by recording the cot over every dimenion at each node of the graph. A in cot-regular, thi could eaily be adapted for filtering by converting thee cot label a cot variable but it would make the algorithm memory expenive. Intead, we invetigate a Lagrangian relaxation approach, which can alo eaily be adapted for filtering from the cot-regular algorithm without memory overhead. 3 A Lagrangian Relaxation-Baed Filtering Algorithm Sellmann [4] laid the foundation for uing the Lagrangian relaxation of a linear program to provide a cot-baed filtering for a minimization or maximization contraint. We apply thi principle to the RCSPP/RCLPP for filtering multicot-regular. The reulting algorithm i a imple iterative cheme where filtering i performed by cot-regular on Π n for different aggregated cot function. In thi ection, we preent the uual Lagrangian relaxation model for the RCSPP and explain how to olve it uing a ubgradient algorithm. Then, we how how to adapt it for filtering multicot-regular. Lagrangian Relaxation for the RCSPP. Conider a directed graph G = (V, E, c) with ource and ink t, and reource (R 1,..., R R ). For each reource 1 r R, let z r (rep. z r ) denote the maximum (rep. minimum 4 ) capacity of a path over the reource r, and c r ij denote the conumption of reource r on 4 in the original definition of RCSPP, there i no lower bound on the capacity: z r

8 arc (i, j) E. A binary linear programming formulation for the RCSPP i a follow: min c ij x ij (1) (i,j) E.t. z r (i,j) E c r ijx ij z r r [1..R] (2) x ij 1 if i =, x ji = 1 if i = t, j V j V 0 otherwie. i V (3) x ij {0, 1} (i, j) E. (4) In thi model, a binary deciion variable x ij define whether arc (i, j) belong to a olution path. Contraint (2) are the reource contraint and Contraint (3) are the uual path contraint. Lagrangian relaxation conit in dropping complicating contraint and adding them to the objective function with a violation penalty cot u 0, called the Lagrangian multiplier. The reulting program i called the Lagrangian ubproblem with parameter u and it i a relaxation of the original problem. Solving the Lagrangian dual i to find the multiplier u 0 which give the bet relaxation, i.e. the maximal lower bound. The complicating contraint of the RCSPP are the 2R reource contraint (2). Indeed, relaxing thee contraint lead to a hortet path problem, that can be olved in polynomial time. Let P denote the et of olution x {0, 1} E atifying Contraint (3). P define the et of path from to t in G. The Lagrangian ubproblem with given multiplier u = (u, u + ) R 2R + i: SP (u) : f(u) = min cx + R R u r +(c r x z r ) u r (c r x z r ) (5) x P r=1 r=1 An optimal olution x u for SP (u) i then a hortet path in graph G(u) = (V, E, c(u)) where: c(u) = c+ R (u r + u r )c r, κ u = r=1 n (u r z r u r +z r ) and f(u) = c(u)x u +κ u. (6) r=1 Solving the Lagrangian Dual. The Lagrangian dual problem i to find the bet lower bound f(u), i.e. to maximize the piecewie linear concave function f: LD : f LD = max f(u) (7) u R 2R + Several algorithm exit to olve the Lagrangian dual. In our approach, we conider the ubgradient algorithm [12] a it i rather eay to implement and it doe not require the ue of a linear olver. The ubgradient algorithm iteratively olve

9 one ubproblem SP (u) for different value of u. Starting from an arbitrary value, the poition u i updated at each iteration by moving in the direction of a upergradient Γ of f with a given tep length µ: u p+1 = max{u p +µ p Γ (u p ), 0}. There exit many way to chooe the tep length for guaranteeing the convergence of the ubgradient algorithm toward f LD (ee e.g. [13]). In our implementation, we ue a tandard tep length µ p = µ 0 ɛ p with µ 0 and ɛ < 1 ufficiently large (we have empirically fixed µ 0 = 10 and ɛ = 0.8). For the upergradient, olving SP (u) return an optimum olution x u P and Γ (u) i computed a: Γ (u) = ((c r x u z r ) r [1..R], (z r c r x u ) r [1..R] ). From Lagrangian Relaxation to Filtering. The key idea of Lagrangian relaxation-baed filtering, a tated in [4], i that if a value i proved to be inconitent in at leat one Lagrangian ubproblem then it i inconitent in the original problem: Theorem 1. (i) Let P be a minimization linear program with optimum value f +, z + be an upper bound for P, and SP (u) be any Lagrangian ubproblem of P, with optimum value f(u) +. If f(u) > z then f > z. (ii) Let x be a variable of P and v a value in it domain. Conider P x=v (rep. SP (u) x=v ) the retriction of P (rep. SP (u)) to the et of olution atifying x = v and let f x=v + (rep. f(u) x=v + ) it optimum value. If f(u) x=v > z then f x=v > z. Proof. Statement (i) of Theorem 1 i traightforward, ince SP (u) i a relaxation for P, then f(u) f. Statement (ii) arie from (i) and from the fact that, adding a contraint x = v within P and applying Lagrangian relaxation, or applying Lagrangian relaxation and then adding contraint x = v to each ubproblem, reult in the ame formulation. The mapping between multicot-regular(x, Z, Π, c), with Z = R+1 and an intance of the RCSPP (rep. RCLPP) i a follow: We ingle out one cot variable, for intance z 0, and create R reource, one for each other cot variable. The graph G = (Π n, c 0 ) i conidered. A feaible olution of the RCSPP (rep. RCLPP) i a path in Π n from the ource (in layer 0) to a ink (in layer n) that conume on each reource 1 r R i at leat z r and at mot z r. Furthermore, we want to enforce an upper bound z 0 on the minimal value for the RCSPP (rep. a lower bound z 0 on the maximal value for the RCLPP). The arc of Π n are in one-to-one correpondance with the binary variable in the linear model of thee two intance. Conider a Lagrangian ubproblem SP (u) of the RCSPP intance (the approach i ymmetric for the maximization intance of RCLPP). We how that a light modification of the cot-regular algorithm allow to olve SP (u) but alo to prune arc of Π n according to Theorem 1 and to hrink the lower bound z 0. The algorithm tart by updating the cot on the graph Π n with c 0 (u), a defined in (6) and then by computing, at each node (i, q), the hortet path k iq from layer 0 and the hortet path k + iq to layer n. We get the optimum value

10 f(u) = k κ u. A it i a lower bound for z 0, one can eventually update thi lower bound a z 0 = max{f(u), z 0 }. Then, by a traveral of Π n, we remove each arc ((i 1, q), a, (i, q )) δ i uch that k (i 1)q + c0 (u) ia + k + iq > z0 κ u. The global filtering algorithm we developed for multicot-regular i a follow: tarting from u = 0, a ubgradient algorithm guide the choice of the Lagrangian ubproblem to which the above cot-filtering algorithm i applied. The number of iteration for the ubgradient algorithm i limited to 20 (it uually terminate far before). The ubgradient algorithm i firt applied to the minimization problem (RCSPP) then to the maximization problem (RCLPP). A a final tep, we run the original cot-regular algorithm on each of the cot variable to hrink their bound (by the way, it could deduce new arc to filter, but it did not happen in our experiment). Note that due to the parameter dependancy of the ubgradient algorithm, the propagation algorithm i not monotonic. 4 Modelling Peronnel Scheduling Problem In thi ection we how how to model tandard work regulation ariing in Peronnel Scheduling Problem (PSP) a one intance of the multicot-regular contraint. The purpoe i to emphaize the eae of modelling with uch a contraint and alo to derive a ytematic way of modelling PSP. 4.1 Standard work regulation In PSP, many kind of work regulation can be encountered, however, we can categorize mot of them a rule enforcing either regular pattern, fixed cardinalitie or liding cardinalitie. To illutrate thoe categorie, we conider a 7 day chedule and 3 activitie: night hift (N), day hift (D) or ret hift (R). For example: R R D D N R D Regular pattern can be modelled directly a a DFA. For intance the rule a night hift i followed by a ret i depicted in Figure 4 (A). The rule can either be given a forbidden pattern or allowed pattern. In the firt cae, one jut need to build the complement automaton. R, D N 1 R (A) a regular pattern example. R, N R, N R, N R, N D 1 D 2 D 3 (B) cardinality rule example. Fig. 4. Example of automata repreenting work regulation.

11 Fixed cardinality rule bound the number of occurrence of an activity or a et of activitie over a fixed ubequence of time lot. Such a rule can be modelled within an automaton or uing counter. For example, the rule at leat 1 and at mot 3 day hift each week can eaily be modelled a the DFA depicted in Figure 4 (B). Taking a look at thi automaton, we can ee the initial tate ha been plit into 3 different tate that repreent the maximum number of D tranition that can be taken. Such a formulation can be an iue when the maximum occurrence number increae. In thi cae, uing a counter i more uitable, a we only need to create a new cot variable z r [1, 3] with c r ij = 1 1 i 7 and j = D. More generally, one can alo encounter cardinality rule over pattern. Thi alo can be managed by mean of a cot. One ha to iolate the pattern within the automaton decribing all the feaible chedule, then to price tranition entering it to 1. Sliding contraint can be modelled a a DFA uing the reformulation tated in [6]. However, the width of the liding equence hould not be too large a the reformulation require to explicit all the feaible tuple of the contraint to lide. Thi i often the cae in PSP or alo in car equencing problem. 4.2 Sytematic multicot-regular Generation A formalim to decribe Peronnel Scheduling Problem ha been propoed in [14]. The et of predefined XML markup allow to pecify a large cope of PSP. In order to automatically generate a CP model baed on multicot-regular from uch pecification, we developed a framework capable of interpreting thoe XML file. In a firt tep, we bounded each markup aociated to a work regulation to one of the 3 categorie decribed above. Hence, for each rule of a given PSP intance, we automatically generate either an automaton or a counter depending on the rule category. For intance, the forbidden pattern no day hift jut after a night hift i defined in the xml file a <Pattern weight="1350"><shift>n</shift><shift>d</shift></pattern> and i automatically turned into it equivalent regular expreion (D N R) ND(D N R). We ue a java library for automata 5 in order to create a DFA from a regular expreion and to operate on the et of generated DFA. We ue the oppoite, the interection and the minimization operation to build an unique DFA. Once the DFA i built, we treat the rule that engender counter, and generate a multicot-regular intance for each employee. Lat, we treat the tranveral contraint and include them in the CP model. For example, cover requirement are turned into global-cardinality contraint. Note that we were not able to deal with two kind of pecification: ome rule violation penaltie that the multicot-regular cannot model and the pattern cardinality rule that we do not yet know how to automatize the reformulation. 5

12 4.3 Two Peronnel Scheduling Cae We firt tackled the GPot [14] problem. Thi PSP conit in building a valid chedule of 28 day for eight employee. Each day, an employee ha to be aigned to a Day, Night, or Ret Shift. Each employee i bound to a (Fulltime or Parttime) contract defining regular pattern and cardinality rule. Regular pattern rule are: free day period hould lat at leat two day, conecutive working week-end are limited and given hift equence are not allowed. Uing the automatic modelling method we preented earlier, we build a DFA for each kind of contract. Cardinality rule are: a maximum number of worked day in the 28 day period i to be worked, the amount of certain hift in a chedule i limited and the number of working day per week i bounded. Cover requirement and employee availabilitie are alo modelled. The oftne pecification on rule ha been ignored a well a the firt pattern rule to avoid infeaibility. The econd cae tudy i baed on the generated benchmark et brought by Demaey et al. [3]. The work regulation arie from a real-world peronnel cheduling problem. The goal i to build only one chedule for a day coniting of 96 fifteen minute time lot. Each lot i aigned either a working activity, a break, a lunch or a ret. Each poible aignment carrie a given cot. The purpoe i to find a chedule of minimum cot meeting all the work regulation. A for the previou PSP, we can identify regular pattern work regulation: A working activity lat at leat 1 hour, Different work activitie are eparated by a break or a lunch, Break, lunch and ret hift cannot be conecutive, Ret hift are at the beginning or at the end of the day, and A break lat 15 minute. And alo fixed cardinality regulation with: At leat 1 and at mot 2 break a day, At mot one lunch a day and Between 3 and 8 hour of work activitie a day. In addition to thoe work regulation, ome activitie are not allowed to be performed during ome period. Thee rule are trivial to model with unary contraint. 5 Experiment Experimentation were run on an Intel Core 2 Duo 2Ghz proceor with 2048MB of RAM running OS X. The two PSP problem were olved uing the Java contraint library CHOCO with default value election heuritic min value. 5.1 On the Size of the Automaton A explained in Section 2.1, the filtering algorithm complexity of the regular contraint depend on the ize of the pecified automaton. Thu it would eem natural that proceing a big automaton i not a good idea. However, practical reult point out two important fact. Firt of all, the operation we run for automatically building a DFA from everal rule tend to generate partially unfolded DFA (by interection) and to reduce the number of redundant tate which lie in the ame layer (by minimization). Hence, the unfolded automaton

13 generated during the forward phae at the initialization of the contraint can even be maller than the pecified automaton Π. Secondly, pruning during the backward phae may produce an even maller automaton Π n a many accepting tate cannot be reached in a given number n of tranition. Table 1 how the number of node and arc of the different automata during the contruction of the multicot-regular contraint for the GPot problem: the um of the DFA generated for each rule, the DFA Π after interection and minimization, the unfolded DFA after the forward and backward phae. Contract Count um of DFA Π Forward Backward (Π n) # Node Fulltime # Arc # Node Parttime # Arc Table 1. Illutration of graph reduction during preolving. 5.2 Comparative Experiment The previou ection howed the eae of modelling with multicot-regular. However, there would be no point in defining uch a contraint if the olving wa badly impacted. We then conduct experiment for comparing our algorithm with a decompoed model coniting of a regular (or cot-regular for optimization) channeled to a global-cardinality contraint (gcc). Table 2 preent the computational reult on the GPot intance. The model include 8 multicot-regular or 8 regular and gcc (for each employee) bound together by 28 tranveral gcc (for each day). In the Table, the firt row correpond to the problem without the liding rule over the maximum number of conecutive working week-end. In the econd row, thi contraint wa included. We tried variou variable election heuritic but found out aigning variable along the day gave the bet reult a it allow the contraint olver to deal with the tranveral gcc more efficiently. Both model lead to the ame multicot-regular regular gcc WE regulation Time () # Fail Time () # Fail no ye Table 2. GPot problem reult olution. Actually, the average time pent on each node i much bigger uing multicot-regular. However, due to better filtering capabilitie, the ize of the earch tree and the runtime to find a feaible olution are ignificantly decreaed.

14 Our econd experiment teted the calability of multicot-regular (MCR) againt cot-regular gcc (CR) on the optimization problem defined in Section 4.3. The model do not contain any other contraint. However the decompoed model CR require additional channeling variable. Table 3 preent the reult on a benchmark et made of 110 intance. The number n of working activitie varie between 1 and 50. The aignment cot were randomly generated. We teted different variable election heuritic and kept the bet one for each model. Note that the reult of the CR model are more impacted by the heuritic. The firt column in the table how that with the MCR model, we were able to olve all intance (Column #) in le than 15 econd for the bigget one (Column t). The average number of backtrack (Column bt) remain table and low a n increae. On the contrary the CR model i impacted a lot a hown in the next column. Indeed, a the initial underlying graph become bigger it contain more and more path violating the cardinality contraint. Thoe path are not dicarded by cot-regular. Some intance with more than 8 activitie could not be olved within the given 30 minute (Column #). Conidering only olved intance, the running time (t) and the number of backtrack (bt) are alway much higher than the MCR model baed reult. Regarding unolved intance, the bet found olution within 30 minute i rarely optimal (Column # opt), and the average gap (Column ) i up to 6% for 40 activitie. MCR model CR model Solved Solved Time out n # t bt # t bt # # opt % % % % % % % 1 Table 3. Shift generation reult 6 Concluion In thi paper, we introduce the multicot-regular global contraint and provide a imple implementation of Lagrangian relaxation-baed filtering for it. Experimentation on benchmark intance of peronnel cheduling problem how

15 the efficiency and the calability of thi contraint compared to a decompoed model dealing with pattern requirement and cardinality requirement eparately. Furthermore, we invetigate a ytematic way to build an intance of multicot-regular from a given et of tandard work regulation. In future work, we ought to get a fully ytematic ytem linked to the CHOCO olver for modelling and olving a larger variety of peronnel cheduling and rotering problem. Acknowledgement We thank Mat Carlon for pointing out the example illutrating the propagation iue of the cot-regular algorithm. We alo thank Chritian Schulte for it inightful comment on the paper and it numerou idea to improve the contraint. Reference 1. Peant, G.: A regular language memberhip contraint for finite equence of variable. In: Proceeding of CP (2004) van Hoeve, W.J., Peant, G., Roueau, L.M.: On global warming: Flow-baed oft global contraint. J. Heuritic 12(4-5) (2006) Demaey, S., Peant, G., Roueau, L.M.: A cot-regular baed hybrid column generation approach. Contraint 11(4) (2006) Sellmann, M.: Theoretical foundation of CP-baed lagrangian relaxation. Principle and Practice of Contraint Programming CP 2004 (2004) Beldiceanu, N., Carlon, M., Debruyne, R., Petit, T.: Reformulation of Global Contraint Baed on Contraint Checker. Contraint 10(3) (2005) 6. Beière, C., Hebrard, E., Hnich, B., Kiziltan, Z., Quimper, C.G., Walh, T.: Reformulating global contraint: The SLIDE and REGULAR contraint. In: SARA. (2007) Maher, M., Narodytka, N., Quimper, C.G., Walh, T.: Flow-Baed Propagator for the equence and Related Global Contraint. In: Proceeding of CP Volume 5202 of LNCS. (2008) Kadioglu, S., Sellmann, M.: Efficient context-free grammar contraint. In: AAAI. (2008) Trick, M.: A dynamic programming approach for conitency and propagation for knapack contraint (2001) 10. Régin, J.C., Puget, J.F.: A filtering algorithm for global equencing contraint. In: CP. (1997) Handler, G., Zang, I.: A dual algorithm for the retricted hortet path problem. Network 10 (1980) Shor, N., Kiwiel, K., Ruzcayǹki, A.: Minimization method for non-differentiable function. Springer-Verlag New York, Inc. (1985) 13. Boyd, S., Xiao, L., Mutapcic, A.: Subgradient method. lecture note of EE392o, Stanford Univerity, Autumn Quarter 2004 (2003) 14. Peronnel Scheduling Data Set and Benchmark: