2 Harald Meyer Auf

ion finding a region searching the region green green green blue blue blue green green blue blue blue green find a new solution that complies with the constraints. the proposed region contains four solutions. red red red red x1 x2 x3 x4 x5 x6 x7 x1 x2 x3 x4 x5 x6 x7 red blue red green green green red blue blue red red green green green red blue blue red red blue red green green green red blue blue red red blue preliminary solution original constraint problem red red red red red red red red red red red red red 0 0 1 0 0 1 abstract constraint problem 1 1 0 1 1 0 1 1 0 0 0 1 rev( x1) rev( x2) rev( x3) rev( x4) rev( x5) rev( x6) rev( x7) 0 0 1 1 0 0 1 solution to the abstraction (avoid to assign 1’s if possible) rev( x1) rev( x2) rev( x3) rev( x4) rev( x5) rev( x6) rev( x7) Figure 6. An example for building an abstra tion of lo al revisions that indi ates bad regions. the extensions of the spe i ed onstraints into a ount. Hen e, this method is able to deal with large arity onstraints, e.g. global onstraints. Beause of its de larative representation as a CSP, any sear h algorithm may be used to derive information on possible improvements from the Boolean abstra tion. NURSE ROSTERING AS CONSTRAINT SATISFACTION : : : 19 After presenting a small theory on lo al and global revisions, an eÆient implementation for the lo al revision of the A hieveSum onstraint is des ribed as an example for the provision of de larative sear h ontrol knowledge in onstraint libraries. Finally, this se tion des ribes the kind of sear h for global revisions that has been used in our ben hmark tests on nurse rostering problems. 4.1. A Small Theory on Lo al and Global Revisions. In the following, rev(v) is de ned to map onstraint variable v of the original problem to the orresponding variable in the Boolean abstra tion. rev(X 0) = frev(x) j x 2 X 0g maps a whole set of variables in the original problem to the pertaining variables in the abstra tion. In order to reason about bad regions in a omplete labeling, we require a formal des ription of regions by di {tuples a ording to Fig. 6. Definition 4.1. Let A1 and A2 be assignments to the variables in X 0. Then di (A1; A2) denotes an assignment to the variables in rev(X 0) su h that 8x 2 X 0 : di (A1; A2) #rev(v)= 0 i A1 #x= A2 #x 1 otherwise. Hen e, di (A1; A2) denotes a 0/1-tuple where a 1 indi ates variables with di erent assignments in A1 and A2. These di {tuples are used to represent global revisions. Definition 4.2. The global revision set of a partial solution A is de ned as fdi (A; A0) j VP(A) VP(A0)g: Consequently, ea h di {tuple in a global revision set indi ates a possible global improvement in the quality of the urrent solution A. The information that we use to derive bad regions omes from an abstra tion of the onstraints in the original problem | the lo al revision onstraint. Definition 4.3. Let A be an assignment to the variables in X and be a onstraint in the original problem. rev( ;A) = hrev(X ); h rev( ;A); f0; 1grev(X )ii is alled a lo al revision onstraint of A respe ting i its extension is fuzzy in the set of all 0/1-tuples over rev(X ) and the membership fun tion rev( ;X) has the property rev( ;X)(ABoolean) = maxf (A0) j di (A0 # X ; A # X ) = ABooleang: The fuzzy extension of lo al revision onstraints maps possible hanges of A | as represented by di {tuples | to their e e t on the degree of satisfa tion of the orresponding onstraint in the original problem. The de nition of rev( ;A) guarantees that this e e t is estimated optimisti ally: The best labeling that omplies with a di {tuple determines its membership. 20 HARALD MEYER AUF'M HOFE Corollary 4.3.1. Forall assignments A0 to the variables in X the following holds true: (A0) rev( ;A)(di (A #X ; A0)): Corollary 4.3.2. The di {tuple di (A #X ; A #X ) that indi ates no hange is a rev( ;A)(di (A #X ; A #X )) = (A #X ) member of the lo al revision onstraint. In the following, rev(C 0; A) is shorthand for mapping a whole set of onstraints of the original problem to their lo al revision onstraints. The above-given de nition allows the dire t onstru tion of revision onstraints for original onstraints that have a small extension, e.g. binary onstraints. Constraints of larger arity, e.g. global onstraints, will mostly require an implementation of their revision onstraint that is taken from a onstraint library. Se tion 4.2 gives an example for su h an implementation. Now, all the information of lo al s ope that is available from ea h onstraint in the onstraint problem has to be integrated into a more omprehensive view on the whole problem. Therefore, we onstru t an abstra tion of the original problem from the lo al revision onstraints that pertain to the onstraints of the original problem. This abstra tion is proper if the valuation of the lo al revisions is similar to the valuation of the onstraints in the original problem. Definition 4.4. The Boolean VCSP rev(P; A) = hrev(V ); f0; 1g; rev(C;A); S; 'i is a proper abstra tion of the VCSP P = hV; D; C; S; '0i i '(rev( ;A)) = '0( ) for all onstraints in C. Remark 4.4.1. A ording to this de nition, a Boolean FHCSP is a proper abstra tion of another FHCSP, if a lo al revision rev( ;A) has the same weight and the same hierar hy level as onstraint . The main result of this se tion is that all global revision sets an be found sear hing the proper abstra tion of the original problem. The following proposition des ribes the relation between solutions in the abstra t problem and solutions in the original problem. Theorem 4.5. Let P be a risp VCSP or a FHCSP and rev(P; A) a proper abstra tion of P. If an assignment A0 to the variables in P is better than the assignment A, then assignment di (A; A0) to the variables in rev(P; A) is better than the assignment di (A; A)). All elements of the global revision set violate less important onstraints in the proper abstra tion than the assignment that assigns a 0 to all variables and, thus, indi ates no hange. Hen e, all global revisions that improve a preliminary solution of the original problem an be found sear hing the proper abstra tion. This method enables, in theory, even the sear h for an optimal solution of the original problem. NURSE ROSTERING AS CONSTRAINT SATISFACTION : : : 21 Proof. Let A00 be the best assignment pertaining to the original problem with di (A; A00) = di (A; A0). Then, the following holds true: VP(A) VP(A0) VP(A00): Due to Corollary 4.3.2, the onstraints in C ontribute with the same membership to VP(A) as their lo al revisions ontribute to Vrev(P;A)(di (A; A)). A ording to Def. 4.3, the onstraints in C ontribute with the same membership to VP(A00) as their lo al revisions ontribute to Vrev(P;A)(di (A; A00)). With Def. 4.4 of proper abstra tions follows VP(A) = Vrev(P;A)(di (A; A)) VP(A00) = Vrev(P;A)(di (A; A0)): Please note, that memberships o ur in fuzzy HCSPs as well as in risp VCSPs, where they have been used to indi ate whether a onstraint is satised or not. Hen e, this proof overs both lasses of onstraint problems. 4.2. Implementing Lo al Revisions. As mentioned above, lo al revisions of onstraints with an expli itly represented extension an be omputed automati ally a ording to Def. 4.3. Controlling the sear h in the original problem by sear hing the proper abstra tion has the advantage, that a onstraint library an provide eÆ ient implementations of lo al revisions for the built-in onstraint types. This advantage is illustrated here by taking a lo al revision of the previously des ribed A hieveSum onstraint as an example. Remember: A hieveSum has a fun tion f as a parameter that maps the values in the domains of the lo al variables to a weight. The sum of the weights has to ome as near as possible to the goal sumgoal. Let ABoolean be an assignment to the variables in the proper abstra tion, fmin[x℄ the minimal weight of a value in the domain of variable v, and fmax[x℄ be the maximal weight of a value pertaining to x. Note: In ontrast to the propagation in the original problem also pruned values may ontribute to fmin and fmax. However, fmin and fmax depend on whether the sear h of the proper abstra tion onsiders a hange in x or not. fmin[x℄ = minff(d) j d 2 Dg if rev(x) 1 is remaining f(A #x) otherwise fmax[x℄ = maxff(d) j d 2 Dg if rev(x) 1 is remaining f(A #x) otherwise A goal sum in between the interval [summin; summax℄ is onsidered to be rea hable by a revision a ording to ABoolean where summin and summax are given as follows: summin(ABoolean) = X x2XA hieveSum fmin[x℄: summax(ABoolean) = X x2XA hieveSum fmax[x℄: 22 HARALD MEYER AUF'M HOFE Consequently, ABoolean is onsidered to have the following membership to rev(A hieveSum;A): rev(A hieveSum;A)(ABoolean) = >>><>>>: sumgoal summax sumtotal if sumgoal > summax summin sumgoal sumtotal if sumgoal < summin 1:0 otherwise This de nition of the onstraint is obviously only an approximation of the orre t lo al revision | one may say, that this is an intervalonsistent approximation. Nevertheless, it leads to an eÆ ient implementation. Again, propagation is done in two steps. For ea h variable, the system omputes fmin[x℄, fmax[x℄. Additionally, the minimal respe tively maximal weights falt max[x℄ and falt max[x℄ are a quired, that refers to one of the assignments ex ept A #x. Hen e, falt max[x℄ and falt max[x℄ represent the e e t that a hange of the assignment to x an have. In the se ond step, the estimate on the best membership of the values 0 and 1 is determined as follows: In ase of summax < sumgoal, the onstraint prefers to ome as near as possible to summax. Value 0 is labeled with a membership of summax fmax[x℄ + f(A #x) sumtotal : Value 1 is labeled with the best e e t a hange an have: summax fmax[x℄ + falt max[x℄ sumtotal : In ase of summin > sumgoal, the onstraint prefers to ome as near as possible to summin. Value 0 is labeled with a membership of summin fmin[x℄ + f(A #x) sumtotal : Value 1 is labeled with the best e e t a hange an have: summin fmin[x℄ + falt min[x℄ sumtotal : Otherwise, the goal sum is onsidered to be a hievable by any kind of lo al revision. 0 and 1 are labeled with membership 1:0. 4.3. Heuristi Enumeration of Global Revisions. The basi idea for applying these results is to sear h the original problem by algorithm iterative-improvement a ording to Fig. 3 ontrolled by a sear h of the proper abstra tion. On smaller problems, the proper abstra tion an be searhed by the bran h&bound. This te hnique alled enumeration of global revisions (EGR) is guaranteed to terminate and an prove the optimality of a returned solution [13, 14℄. However, the large size of nurse rostering NURSE ROSTERING AS CONSTRAINT SATISFACTION : : : 23 hoose-bad-region(X, C, , l, Æ, sizemax) 1. if s hedulerX has no value (or has been deleted) then load all variables, whi h are lo al to a onstraint with (A) < 1 into s hedulerX ; 2. if s hedulerX 6= fg then begin (a) hoose a x 2 s hedulerX ; (b) set s hedulerX := s hedulerX n fxg; ( ) return X 0 = fxg and exit; end 3. if A has been improved by the previous improvement step of iterative-improvement, then delete s hedulerX and goto 1; 4. Prev := rev(P ; A); onstrain the solutions of Prev to assign at most a number of sizemax 1's; 5. all bran h&bound on Prev for the rst omplete labeling ABoolean that omplies with all hard onstraints; if no solution has been found return X 0 = fg and exit; 6. as long as ABoolean violates too important onstraints in order to promise an improvement of A repeat begin (a) if time is running out then return X 0 = fg and exit; (b) olle t all variables that ABoolean labels with a 1 into set X 00; ( ) hoose randomly sizemax additional variables and add them to X 00 (d) ABoolean = optimize region(Prev ; X 00; ABoolean); end; 7. return X 0 = fx j ABoolean #rev(x)= 1g. Figure 7. Heuristi enumerating global revisions (HEGR). problems requires a lo al sear h strategy also when sear hing for global revisions at the ost of loosing the properties of guaranteed termination and proving optimality. Hen e, this te hnique is alled heuristi EGR (HEGR). Figure 7 sket hes the implementation of hoose-bad-region a ording to the HEGR prin iple. One key problem of sear hing for global revisions is that all variables in the proper abstra tion have to be labeled for generating a rst result. To over ome this problem, hoose-bad-region follows themin on{ strategy [15, 21℄ as long as this is useful. Therefore, s hedulerX is loaded with all variables that ontribute to a on i t. As long as this variable set is not empty, the algorithm re ommends to hange only a single variable assignment (row 2). If all variables, that have been loaded into s hedulerX , have been tried, the algorithm assumes that the urrent assignment in the original problem A is a Pareto-optimum that annot be improved this way. This state is rea hed in row 4 of Fig. 7. From here on, the proper abstra tion is sear hed for a global revision. On larger problems, this still annot be done by a pure tree-sear h algorithm. This version of hoose-bad-region follows an iterative-improvement 24 HARALD MEYER AUF'M HOFE strategy. However, as a prerequisite, the size of the bad regions to return has to be onstrained by a parameter sizemax. In the experiments as they are des ribed in the next se tion, we use a region size of 8. On the one hand, this value for the region size promises s hedules of a reasonable quality be ause we assume lo al minima that resist the hange of 8 assignments as very rare. On the other hand, this value ensures an a eptable run-time behavior of the improvement step in optimize-region, that is ondu ted by an exhaustive bran h&bound sear h. Pro edure hoose-bad-region produ es an initial assignment to all variables in the proper abstra tion in row 5 of Fig. 7. This assignment may violate too many onstraints in order to be useful. Remember: A ording to Theorem 4.5, ABoolean has to be better than di (A; A). As long as ABoolean is too bad, hoose-bad-region ondu ts an iterative-improvement-pro edure in the loop in row 6. The advantage to the iterative-improvement of the original problem is that we know mu h more about Prev than we know about P. The number of 1's in a solution of Prev is xed. Hen e, the only opportunity to improve ABoolean is to repla e a rev(x) 1 by a better re ommendation for a hange to the original problem. X 00 de nes the region that is optimized in Prev. This variable set omprises all variables, that are urrently labeled with 1, and as many randomly hosen variables, that are labeled with a 0. Pro edure optimize-region is then used to test, whether it is better to hange the distribution of 1's on the variables in X 00. This loop is left if a promising region has been found. The on luding improvement step in iterative-improvement still may fail, although the merit of ABoolean promised to represent a global revision. Su ess or fail of the next improvement step has some in uen e on the all of hoose-bad-region, that follows this improvement step. If the previous improvement step has been a su ess then hoose-bad-region returns into the min on{mode. Otherwise, the pro edure pro eeds in urrent mode. This is ontrolled in row 3. 5. Experiments This se tion presents the results of some experiments on nurse rostering problems that use the new fuzzy onstraints. The ben hmark problems do not refer to ertain working time models in order to abstra t from the e e t of lever or unsuited models. The onstraint hierar hy is organized as follows: 1: At least a 10 hours break is required between two working shifts. Ea h violation is weighted with 1.0. One early-morning-shift, late-shift, and night-shift has to be served on ea h day as minimal attendan e. For ea h day and ea h attendan e spe i ation this onstraint is weighted with 1.0. At most 10 working shifts an be served by the same employee within a fortnight. For ea h employee, this onstraint has weight 1.0. All of these onstraints are ompulsory but have been NURSE ROSTERING AS CONSTRAINT SATISFACTION : : : 25 6.00am 2.00pm 10.00pm 6.00am day-turn 1 2 3 early-morning 1 2 3 late-shift 1 2 3 night-shift 1 2 3 Figure 8. Start, end, and duration of working shifts. moved into hierar hy level one. Hen e, the s heduling pro edure is not for ed to deliver legal s hedules. 2: Management of working time a ounts and standard rew attendan e is done in hierar hy level 2. The onstraint weights are hosen in su h a way that oming one person's attendan e nearer to the standard rew is worth four hours overtime work. Su h a representation requires the use of fuzzy onstraints and has, thus, not been used in the ommer ial system. The ommer ial system puts working time management and onstraints on the standard rew into di erent hierar hy levels. 3: A break of 14 hours is preferred in hierar hy level 3. 4: The attendan e of rew members is preferred not to ex eed the standard rew size. This model still negle ts holidays and several onstraints on sequen es of shifts, that are part of the ommer ial system, but it draws a suÆ iently good pi ture of nurse rostering problems. The ben hmarks use two sets of working shifts a ording to Fig. 8. The small set omprises morning-shift 1, late-shift 1, night-shift 1, and the three day-turns. The larger set omprises all displayed shifts. The larger set of shifts, whenever unrealisti , has been spe i ed to make the relation between working time and rew requirements more omplex but also more exible. All problems exhibit the same onstraints on rew attendan e. The Figures 9 to 11 show the results of three algorithms. The urves present the weight of onstraint violations of the best yet found solution on a time axis. The following algorithms have been used: min on-walk 0.02 (light gray solid urve) uses the low walk fa tor 0.02. The initial labeling is found using forward he king and the most signi ant value heuristi for dynami variable ordering. min on-walk 0.2 (bla k dots) is similar to min on 0.02 but with a higher walk fa tor of 0.2. The gures show a dot whenever min onwalk has found a lo al improvement. This kind of presentation draws a good pi ture of the dynami behavior if min on-walk with larger walk fa tor. heuristi -egr (dark gray solid urve) is used with sizemax = 8. 26 HARALD MEYER AUF'M HOFE time/s weight of violated constraints mincon-walk 0.2 mincon-walk 0.02 heuristic-egr region size 8 9 nurses. 4 half day contracts, three shifts + 3 day-turns 1e+07 1e+08 1e+09 1e+10 1e+11 1e+12 1e+13 1e+14 0 50