JP7169494B1 - program - Google Patents

Abstract

An object of the present invention is to output a solution with an optimum solution guarantee at high speed. Kind Code: A1 In a sequence generation method, a linear primal problem solver at code 1 outputs a dual to a pricer block at code 11 . The per-staff pricer at 2 creates a ZDD graph encoding the shifts in one-hot code for the hard and soft constraints associated with each staff. Obtain the minimum cost of the ZDD nodes between the start and end points of the ZDD graph and determine whether to add the column with code 10. [Selection drawing] Fig. 6

Description

TECHNICAL FIELD The present invention relates to programs and the like that optimize nurse scheduling problems and the like.

Conventionally, for example, the nurse scheduling problem of optimizing the schedule of staff belonging to a given group is known. The nurse scheduling problem is known as an NP-hard example of constraint satisfaction optimization problem systems. As a solution to this type of scheduling problem, there is a branch and price method, as disclosed in Non-Patent Document 1, Non-Patent Document 2, and the like.

Strandmark, P., Qu, Y. and Curtois, T. First-order linear programming in a column generation-based heuristic approach to the nurse rostering problem. Computers & Operations Research, 2020. 120, p. 104945. (pdf) Antoine Legrain, J.Omer, Samuel Rosat. A rotation-based branch-and-price approach for the nurse scheduling problem. Mathematical Programming Computation, Springer, 2020, 12 (3), pp.417- 450. ff10.1007/s12532 -019-00172-4ff.ffhal-01545421v2 Minato, S. 1993. Zero-suppressed BDDs for set manipulation in combinatorial problems. 30th Conference on Design Automation. 272-277. J. Kawahara, T.; Inoue, H.; Iwashita, and S.; Minato, "Frontier-based search for enumerating all constrained subgraphs with compressed representations", IEICE Transactions on Fundamentals of Electronics, Communications, Communications Corporation. E100. A, no. 9, pp. 1773-1784, 2017. Modeling and solving staff scheduling with partial weighted maxSATEmir Demirovic, Nysret Musliu & Felix Winter Annals of Operations Research volume 275, pages79-99 (2019)

As shown in Non-Patent Document 1, in the nurse scheduling problem, a design approach that attempts to increase the speed of finding a solution results in a decrease in accuracy with respect to the optimal solution. Conversely, in a design aimed at high precision, the solution-finding speed tends to decrease. There is a problem that it is difficult to achieve both high precision and high speed.
SUMMARY OF THE INVENTION In view of such problems, it is an object of the present invention to enable, for example, a solution with an optimum solution guarantee to be output at high speed.

According to a first aspect of the present invention, a program for causing a computer to execute a process related to schedule optimization based on shifts constituting a schedule of staff members belonging to a predetermined group and constraints on the schedule is provided so that the constraints must be observed. It includes hard constraints that should be constrained and soft constraints that do not necessarily need to be met but are desired to be met if possible. By finding the path that minimizes the total cost of the ZDD nodes between the start and end points of the ZDD graph encoded in (zero-suppressed binary decision graph), the hard constraints are satisfied and the weights of the constraints violating the soft constraints are calculated. A first means for obtaining the solution of the optimization problem of each stuff that minimizes the sum, obtains the integer solution based on the sequence generation method in which the solution of the optimization problem of each staff is the sequence, and the branch-and-bound method. It is a program for making a computer function as the second means.

According to the present invention, it can be expected that the ratio of obtaining a solution with the optimum solution guarantee at a high speed will be significantly improved. If the optimum solution is obtained, it is considered that there is no better solution, so the search can be stopped, and the effect of preventing the user from waiting unnecessarily can be expected.

FIG. 1 is a block diagram of a method for solving the nurse scheduling problem with a sequence generation method. Figure 2 is a concrete example of hard and soft constraints Figure 3 is the algorithm for the column generation method. FIG. 4 is a configuration diagram according to the conventional method. FIG. 5 is an explanatory diagram of the branch-and-bound method. FIG. 6 is a configuration diagram of the present invention. Figure 7 shows a day's worth of shifts encoded into ZDD with one-hot code. Figure 8 is a diagram of 5 days worth of shifts encoded into ZDD with one-hot code. FIG. 9 is a diagram in which prohibition of day shift (D)→early shift (E) is added to FIG. Figure 10 is the ZDD graph of Figure 2 implementing all row constraints except the cardinality constraint FIG. 11 is a ZDD graph obtained by adding a cardinality constraint to FIG. FIG. 12 is a ZDD graph with schedule constraints added to FIG. FIG. 13 is an implementation example of the radix constraint by a binary adder. Figure 14 is an implementation example of cardinality constraint by ZDD set Fig. 15 is an example in which the weight of the schedule constraint is given to the ZDD node Fig. 16 is a ZDD graph obtained by soft-constraining Fig. 9 FIG. 17 is a ZDD graph in which FIG. 10 is soft-constrained. FIG. 18 is an explanatory diagram of ZDD cost calculation focusing on one node of ZDD when there is no soft cardinality constraint. FIG. 19 is an explanatory diagram of ZDD cost calculation focusing on one node of ZDD with soft cardinality constraints. FIG. 20 is a block diagram of a method for solving the pricer with sequence generation. FIG. 21 is an explanatory diagram of how to input from the main problem solver of FIG. 6 to the MaxSAT solver. FIG. 22 is an explanatory diagram of converting the soft constraint of the row constraint into a hard constraint using the solution of the MaxSAT solver. FIG. 23 is a speed comparison between the present invention and a mathematical solver FIG. 24 is a comparison graph between the present invention and other methods showing the ratio of the number of instances for which the optimal solution was obtained over time.

An example of an embodiment to which the present invention is applied will be described below with reference to the drawings.
It goes without saying that the embodiments to which the present invention can be applied are not limited to the embodiments described below.

[1. Branch pricing method]
FIG. 1 shows an example of an overall block diagram of a system for obtaining work solutions using the branch pricing method. This block diagram may be a framework that does not change between the conventional method and the present invention. This block diagram will be described below.

[1.1 Hardware/Software Constraints]
Constraints are divided into two types. "Hard Constraints" and "Soft Constraints".
A hard constraint is a constraint that must be observed.
A soft constraint is a constraint that you do not necessarily have to observe, but you want to observe if possible.
A specific example of constraints is shown in FIG.

Constraints can include row constraints, which are constraints mainly related to each staff member, cover constraints, which are constraints for the entire organization, schedule constraints, such as fixed schedules for each staff member, and the like. There is a degree of freedom in deciding which one to use as a hard constraint or a soft constraint. In the case of soft constraints, if they are broken, a cost is generated, which is determined for each soft constraint. For the system as a whole, the total cost of violating the soft constraints while observing all the hard constraints is called the objective function value. Minimizing the objective function value is a requirement for outputting the optimal solution. .

As described above, there is a degree of freedom in deciding which constraint to use as a hard constraint or a soft constraint. As one method, for example, the user may decide (set).
Alternatively, the computer may automatically determine (set) by a program.

[1.2 Branch pricing algorithm]
FIG. 3 shows an example of a flow chart of the branch pricing method. The branch-price method consists of a string generation (string generation block), which is a variant of the simplex method, and a branch-and-bound method (branch-and-bound block).

[1.3 Conventional system]
FIG. 4 is a diagram showing an example of a block diagram of a column generation block of a conventional system.
The column generation block consists of a group of subprograms (hereafter referred to as "Pricer") related to the constraint of each stuff and a linear relaxation primal problem solver for each stuff.
A linear relaxed primal problem solver is a solver that solves a problem that is relaxed from a discrete system to a continuous system. For example, there is an open source COIN-OR project. A linear relaxation primal problem solver outputs two solutions, a primal solution and a dual solution. Pricer outputs a discrete solution (hereinafter referred to as a "roster") for its own schedule. A roaster is introduced as a variable into the primal problem solver as a "string" if it contributes to the improvement of the primal problem system.

On the other hand, if it is determined that it does not contribute, it is not taken in. Columns are added as long as the objective function value of the primal problem continues to improve. The bottom bound of the system is when no more columns are finally added. The lower bound here is the minimum value in a continuous system. The objective function value of the real system of the nurse scheduling problem is a discrete system and generally has the property of being an integer value equal to or always larger than the lower bound.

[1.3.1 Pricer]
Pricer can be generalized to the resource constrained shortest distance problem. Pricer is known to be an NP-hard problem. Pricer's output roster is the key to the accuracy and speed of the overall system. To date, various proposals have been made to solve this problem. As one of them, Non-Patent Document 1 describes a method of obtaining a solution by heuristics.

各シンボルの意味を数６に示す。

[1.3.2 Sequence Generation Block Linear Relaxed Primal Problem Solver]
Each additional column increases the number of variables, but at most the variable increase is only for the columns. Besides, the linear relaxed primal problem solver only deals with the cover constraints between staffs, and achieves the overall optimization with far fewer variables than the entire nurse scheduling problem. be able to. For this reason, the linear relaxation master problem is sometimes called the "restricted master problem". The linear relaxation primal problem solver outputs a dual solution (DUAL) to each pricer, and each pricer receives it and returns the optimal roaster, thereby improving the objective function value of the primal problem.
The objective function of the linear relaxation primal problem can be formulated as Equation 1.

Expression 6 shows the meaning of each symbol.

Equation 5 is Pricer's objective function, also called "Reduced Cost". When the Reduced Cost is negative, the objective function value in Equation 1 has the potential for improvement, so its roaster is added as a column to the linear relaxation primal problem solver. When there is no roaster for which the objective function value of Equation 5 becomes negative, the improvement ends. At this time, it is known that the optimal value of Equation 1 indicates a lower bound. Conversely, in order to correctly know the lower bound of the system of the nurse scheduling problem, it is necessary for the pricer to output a value that is the optimal solution.

[1.4 Branch-and-bound method]
The primal solution output of the linear relaxation primal problem solver is a continuous system and is output in floating point. However, what the system wants is an integer solution, and floating-point data (for example, 0.3 or 0.5) cannot be a solution, so integerization must be performed. The block needed for this is the Branch and Bound Block. The optimal value of the system is obtained by integerization by the branch-and-bound operation while obtaining the lower bound by sequence generation. A node is generated each time a branch occurs, and an integer solution (hereinafter referred to as a "feasible solution") is obtained at the time when all data can be converted to integers. The optimal value is determined when the search for all nodes is completed or when the minimum value of feasible solutions=UB becomes equal to the lower bound value (=LB). For the branching operation, it is common to select a fractional value (for example, 0.5) and branch each fixedly to 0/1. FIG. 5 shows this situation.

[1.5 Heuristics]
Heuristics are sometimes used as a way of finding feasible solutions early on. General heuristics, as described in Non-Patent Document 1, perform integerization by repeatedly rounding floating-point data.

[2. Key points of the present invention]
FIG. 6 is a diagram showing an example of system configuration for obtaining a solution according to the present invention.
Compared with the system configuration shown in FIG. 4, the configuration of reference numeral 2 is characteristic.
Reference numeral 2 is a pricer. Specifically, for hard and soft constraints related to staff, it is characterized by outputting a roaster that optimizes each staff using a zero-suppressed binary decision diagram (ZDD) that encodes shifts with one-hot code. be.
Here, the hard constraint and soft constraint handled by the pricer 2 can be, for example, the row constraint and schedule constraint described above.

One hot code is, for example, early shift: (1, 0, 0, 0), day shift: (0, 1, 0, 0), late shift: (0, 0, 1, 0), public holiday: (0, 0 , 0, 1), there is only one part that is "1", and the positions of the parts are different from each other. FIG. 7 shows this as a ZDD graph. The point is that it can compactly express all possible states of staff shifts.
Under this encoding condition, once we have a graph that encodes the hard and soft constraints on the stuff on the ZDD, we can use it to quickly compute the optimal solution of Pricer. In the sequence generation method, pricer calculation is a bottleneck of the system.

[3. Embodiment Based on Nurse Scheduling Problem]
In this embodiment, Pricer's objective function, for example, Equation 6 described above, is expanded with detailed constraints for each staff. Thus, for example, Equation 7 can be obtained.

[3.1 Implementation of one-hot code]
As an example, we implement the row constraint of FIG. Here, it is assumed that "early shift = E", "late shift = L", and "day shift = D". That's all for the yang shift, but other states, that is, non-working days, are also considered shifts, and this is set as "Public holiday = Y" and constitutes a one-hot code. For example, assuming that Date ranges from "1" (Date1) to "5" (Date5), the earliest date of Date1 is written as E1. Let R be the constraint logic function of a staff.

Since the one-hot code is Least One and Most One, we can implement the OR of all shifts and the constraint that no two shifts are active at the same time.
Using the logical operators & (AND), | (OR), NOT (~), the constraint logic function R is
R=E１ | D１ | L１ | Y１
R &=~ (E1 & D1 )
R &=~ (E1 & L1)
R &=~ (E1 & Y1)
R &=~ (D1 & L1)
R &=~ (D1&Y1)
R&=~(L1&Y1)
can be described as
The above was implemented only for Date1, but other Dates can be constructed in the same way.

To graph this logic function, it is easy to use the BDD package, which is widely used as an LSI logic synthesis tool. A BDD is a Binary Decision Diagram, which can express logic functions compactly. It is known that an irreducible BDD has a unique representation.

As a BDD package, you can use the open source CUDD (Colorado University Decision Diagram Package). BDDs are suitable for representing logical functions. On the other hand, ZDD can be said to be a set representation of BDD, and it is useful when making a data structure a set representation. AND on BDD is ∩ of sets on ZDD. CUDD comes with a converter that converts from BDD representation to ZDD representation, and the above logic function can also be converted to ZDD representation. ZDD and its irreducible representation are unique.

FIG. 8 is a ZDD representation of the constraint logic function R. As shown in FIG. When the complicated logic function above is made into a ZDD graph, it can be seen that it can be expressed very compactly. In FIG. 8, D1 is the starting point of the ZDD graph and T is the ending point.

[3.2. Implementation of Hard Row Constraints]
Next, try to implement the line restriction 20 "prohibit early shift after day shift" in FIG. Performing an AND operation on the previous one-hot code logic function, the constraint logic function R becomes:
R & = ~ (D1 & E2)
R &=~ (D2 & E3)
R &=~ (D3 & E4)
R &=~ (D4 & E5)
FIG. 9 shows the ZDD converted from the constraint logic function R at this point.

Similarly, FIG. 10 shows the implementation of all row constraints other than the cardinality constraint.

[3.2.1. Implementation example of cardinality constraint]
Counting the elements of the shift set and constraining the sum with an inequality is called a cardinality constraint. For example, among the row constraints in FIG. 2, the cardinal number constraint is only the constraint 21 stating that the number of public holidays is between 1 and 2 days.
There are, for example, two approaches to implementing cardinality constraints.

The first method is a method using a binary adder on the BDD, and the circuit synthesis method can be applied as it is. FIG. 13 shows an example of the configuration. A shift set of interest is input to a binary adder. An inequality will constrain the output of the binary adder. Concretely, it is sufficient to constrain using a comparator, which is also a circuit synthesis method. All of the above can be described using only the basic operations of AND, OR, and NOT, which are the primitive elements of the circuit, and can be described using the BDD package CUDD.

The second method is a method based on set operations. For example, the set ``1<=X<=2'' can be considered as the union of all combinations ``nC1'' that take 1 and ``nC2'' that take 2 from the n input set. can. Alternatively, it can be considered as the intersection of the set "1<=X" and the set "X<=2". FIG. 14 shows the latter situation. Regarding cardinality constraints, for example, there is a ZDD package TdZdd (URL: https://github.com/kunisura/TdZdd) that directly drops ZDD from the specification, and it is easy to use. In this case, except for the cardinality constraint, since it was built on BDD, convert from BDD to ZDD. The whole ZDD can be obtained by performing the ∩ operation on the non-radix-constrained ZDD Z1 and the cardinal-constrained Z2. ZDD set operations are also possible on CUDD.
FIG. 11 is the result of compiling with cardinality constraints added. It can be seen from FIG. 8 that there is no new variable increase.

[3.3. Implementation of Hard Scheduling Constraints]
To set the work on January 7 in Figure 2 as a day shift (D2),
R&=D2
And it is sufficient.
Similarly, to set the work on January 8 in Figure 2 as other than the late shift (L3),
R &= ~ L3
And it is sufficient. FIG. 12 is a graph of the constraint logic function R at this point. In this way, compilation of arbitrary schedules, arbitrary hard constraints is possible by encoding shifts into ZDDs in one-hot code.
All of the hard constraints have now been implemented.

[3.4. Compilation Order of Hard Constraints]
No matter what order you compile the ZDD, you will end up with Figure 12. This is because the representation of an irreducible ZDD is uniquely determined. In general, ZDD construction time is proportional to the number of nodes contained in the ZDD graph. Among FIGS. 8 to 12, FIG. 11 has the largest number of nodes, but finally, FIG. 12 is the second smallest graph after FIG. As can be seen from this, it is advantageous in terms of build time and memory requirements to compile the one-hot constraint and the hard schedule constraint first, and then implement the other constraints. Especially, the larger the instance, the greater the effect.

[3.5. Implementation of Soft Scheduling Constraints]
From here on, the description will be continued on the assumption that [3.1 Compilation of one-hot code] is completed.
In the case of soft constraints, adding "if possible" to the constraints in FIG. 2 results in soft constraints. For example:
・Working on January 7th, if possible, day shift (D2)
・Working on January 8th, if possible, other than the late shift (L3) However, "if possible" is a wish, and you have to accept what you can't do. In the case of scheduled constraints, there is no change in the shape of the graph before and after applying the constraints.
In fact, Figure 15, with the soft scheduling constraint applied, is the same as Figure 8 after the one-hot code has been applied. The difference is the weight.
The difference is that a predetermined constraint weight is added to one branch node of ZDD. Assuming that the weight of January 7 is w2 and the weight of January 8 is w3, for example, the scheduled constraint weight of reference numeral 80 in FIG. is w2, the predetermined constraint weight of reference numeral 84 is w3, and the predetermined constraint weight of reference numerals 85-87 is "0".

[3.6. Implementation of Soft Row Constraints]
Next, if possible, implement a ban on the early shift after the day shift. A new variable X is used to represent the prohibited state, and X is weighted with p. In the following, == denotes equivalence.
R & = X1 == (D1 & E2)
R & = X2 == (D2 & E3)
R & = X3 == (D3 & E4)
R & = X4 == (D4 & E5)
FIG. 16 is a ZDD graph of the constraint logic function R. FIG. The branch of code 90-93 may be set as the weight p.

Similarly, FIG. 17 shows the implementation of soft constraints for all row constraints other than cardinality constraints.

[3.7. Shortest path calculation method when soft cardinality constraints are not included]
A calculation method when the soft constraint weight gk in Expression 7 is 0 will be described.
Generalizing the ZDD node of FIG. 17, we obtain FIG. When res is the Boolean variable of the ZDD node indicated by reference numeral 100 and val is the value, res and val are TVal1 of the child node 101, Eval1 of the child node 102, and It can be calculated only with the weight w (symbol 104). The weight w is determined by the weight of the soft scheduled shift in FIG. As for the hard cardinality constraint, as shown in [3.2.1. Implementation example of cardinality constraint], neither variable increase nor weight generation occurs. Therefore, it can be calculated at reference numeral 103.

[3.8. Shortest path calculation method when including soft cardinality constraints]
FIG. 19 shows a calculation example of the shortest path when even one soft cardinality constraint is included. A ZDD node denoted by reference numeral 116 has reference numerals 114 and 115 as child nodes. Reference numeral 115 is connected by one branch. The one-edge case has a weight and a delta term for the cardinality constraint. Here, as an example, the weight w is set to "-1" and the cardinality constraint delta term is set to "1". The total sum of radix constraint delta terms of one branch at each ZDD node up to that point is held in memory as Counts denoted by reference numeral 117 . Counts is the total number of shifts so far. Each ZDD node has a table of Counts and their corresponding weights. The table code 112 of the parent node code 116 is obtained from the 1 branch node code 115 plus the code 113 and the 0 branch node code 114 . A reference numeral 112 performs a merge operation to adopt a smaller COUNTS value for those having the same COUNTS value to obtain a table indicated by reference numeral 111 . At the final node (reference numeral 110), the cardinality constraint weight is further added to obtain the value corresponding to COUNTS. Finally, the smallest value becomes the objective function value of Expression 7, which is the desired path.

In FIG. 19, by scanning from the end node to the start node, the objective function value of Expression 7 can be calculated. The dual solution of the primal problem may change each time, and Equation 7 needs to be recalculated each time. On the other hand, the structure of ZDD nodes is immutable once compiled.
Number 7 will be calculated many times during column generation, but once the ZDD structure is determined, Counts can be calculated once and saved in the form of a table. This makes it possible to increase the speed. Moreover, the code|symbol 117 may be data-compressed and preserve|saved. Conversely, scanning may be performed from the start node to the end node. These procedures find the path that minimizes the sum of the weights of the ZDD nodes between the start and end points. By the method described above, it is possible to obtain an exact solution of Pricer at high speed. Calculation is performed using Equation 7, and if the result is negative, a column is added (reference numeral 10 in FIG. 6) as a column.

[3.9. Another shortest path solution when cardinality constraints are included]
As another solution, it is also possible to solve the cardinality constraint by the sequence generation method and the branch-and-bound method, in which the main problem of the sequence generation method and the sequence generation method that graphs the other than the cardinality constraint with ZDD and makes it a sequence generator problem. [3.8. Shortest Path Calculation Method with Soft Radix Constraints] gives the exact solution of Expression 7, but if the instance size is large, it may be constrained by the installed memory. In that case, memory requirements can be reduced by restricting ZDD compilation to non-cardinality constraints. FIG. 20 is a configuration diagram thereof.
FIG. 20 shows a configuration applicable to each pricer 2 shown in FIG. A block 141 for obtaining the path with the minimum sum of node costs between the start and end points of the ZDD graph whose shift is encoded with a one-hot code for the constraint; and a block 144 for obtaining an integer solution 145 based on the solution 143 by a branch and bound method.

[3.10. Parallelization of Pricer]
In reference numeral 11 in FIG. 6, the pricers corresponding to each stuff are completely independent, and parallelization by a multiprocessor can be easily configured. In particular, the ZDD structure has a sequential access structure, and even if the ZDD graph does not fit in the cache size of the CPU, it can provide excellent access characteristics. As a result, high performance can be maintained even in a multiprocessor configuration. In the multi-processor configuration, if there is room in the number of CPUs, one CPU may be assigned to each staff, or a GPU may be used. If there is no room, physical CPUs can be allocated by scheduling with OPEN-MP or the like. Also, one CPU may be assigned to each group of staff.

MaxSAT (Maximum Satisfaction Assignment Problem) is a truth-or-false problem that maximizes the number of clauses that satisfy a logical variable in a logical expression written in Conjunctive Normal Form (CNF). It is a question of finding a value. Weighted MaxSAT is a truth value assignment that maximizes the sum of the weights of satisfying clauses for variables in a logical formula written in CNF when weights are assigned to the clauses. It is a question to ask. Weighted Partial MaxSAT is an optimization problem considering a clause that must be satisfied (hardware) and a clause that does not have to be satisfied (software) for Weighted MaxSAT described above. Non-Patent Document 5 describes a method of converting a scheduling problem into Weighted Partial MaxSAT and solving it. Here, the Weighted Partial MaxSAT solver is simply referred to as the MaxSAT solver.

[3.11. Integer conversion by MaxSAT solver]
As a method of obtaining a solution at a higher speed, the MaxSAT solver shown in FIG. 6 will be described. The input to the MaxSAT solver is the code 5 in FIG. 6 and the code 5 in FIG. 1, and the specific contents are as shown in FIG. In other words, the constraints of the entire system are converted into CNF and input to the MaxSAT solver. The MaxSAT solver outputs code 9 integer solutions. As a result, the MaxSAT solver alone has the ability to obtain solutions for the entire system.

As the input of this MaxSAT solver, it is characterized in that code 7 integer data among code 1 output of the solver of the linear relaxation main problem, code 6 floating point data is input and taken in. An example of this is shown in FIG. The data of code 6 are generally floating point numbers. The conventional heuristic method performs integer conversion by setting floating point data, for example, 0.9→1, 0.1→0. The data with the highest reliability among them is an integer value, and the data with the lowest reliability is 0.5. Therefore, as the data to be input to the MaxSAT solver, integer data is input as auxiliary data. A constraint propagation mechanism within the MaxSAT solver serves as a mechanism for estimating the as yet undetermined binary data from the input ancillary data and the global constraints of the system. As a result, a high-precision feasible solution is quickly output. This mechanism can also operate during branch-and-bound.

[3.12. Initial solution generation by MaxSAT solver]
The MaxSAT solver is fast in deriving a solution that satisfies the hard constraints (hereinafter called a feasible solution). The MaxSAT solver of FIG. 6 code 4 may be run first and its solution code 9 may be taken as FIG. 3 code 30 initial solution of linear primal problem. In this way, the starting point for improving the main problem can be the objective function value output by the MaxSAT solver, which contributes to speeding up convergence. Also, if the MaxSAT solver's objective function value is below the weight of the soft constraints that make up the ZDD, then the soft constraints can be compiled as hard constraints. An example is shown in FIG. This makes the ZDD structure more compact and achieves even higher speeds.

[4. Effects of the present invention]
A solution with optimal solution guarantee can be obtained at high speed even for large instances. 23 and 24 are comparisons with the state-of-the-art mathematical solvers according to the same instances as in Non-Patent Documents 1 and 2. FIG.
ScheduleNurse3 is the solver to which the present invention is applied. Clearly, it can be seen that the method according to the invention is superior.
Also, the academic benchmarking site (http://www.schedulingbenchmarks.org/changes.html) has international record updates for nurse scheduling problems. The updates by the inventor (Tak. Sugawara) are all the result of this invention and also contribute to solving a long standing problem.

1 linear relaxed primal problem solver 2 pricer 3 branch and bound block 4 MaxSAT solver 5 hard and soft constraints 6 primal solution of sign 1 (floating point output)
7 Integer data output with code 1 8 Integer solution by branch and bound method 9 Integer solution by MaxSAT solver 10 Column addition to primal problem 11 Pricer block 12 Dual solution output of primal problem solver 13 Roaster 20 Row constraint example 21 Cardinality constraint Example 30 Initial solution in sequence generation method 80 Weight of day shift D2 81 Weight of early shift E2 82 Weight of late shift L2 83 Weight of holiday Y2 84 Weight of late shift L3
85 Weight of early shift E3 86 Weight of day shift D3 87 Weight of holiday Y3 90 Weight of day shift (D1) → early shift (E2) 91 Weight of day shift (D2) → early shift (E3) 92 Weight of day shift (D3) → early shift (E4) weight 93 day shift (D4) → early shift (E5) weight 100 ZDD node 101 child node of code 102 child node of code 102 child node of code 103 algorithm for value, boolean determination 104 weight of code 100 110 weight of soft cardinality constraint at final node Result of merging 111 Result of merging child node result 112 Intermediate progress of merging child node result 113 Code 116 Node weight and delta count 114 Child node 115 Child node 116 ZDD node 117 Shift sum 140 Pricer Prime problem solver 141 ZDD other than cardinality constraints in pricer
142 column addition to code 140 143 principal solution of code 140 144 branch and bound block 145 integer solution of code 140 146 dual solution of code 140

Claims (5)

A program for causing a computer to execute processing related to schedule optimization based on shifts constituting a schedule of staff members belonging to a predetermined group and constraints related to the schedule,
The constraints include hard constraints, which are constraints that must always be observed, and soft constraints, which are constraints that do not necessarily need to be satisfied but are desired to be satisfied if possible,
For each staff, for the hard and soft constraints associated with each staff, minimize the sum of the costs of the ZDD nodes between the start and end points of the ZDD graph that encodes the shifts into a ZDD (Zero Suppressed Binary Decision Diagram) with one-hot code. a first means for solving an optimization problem for each staff that satisfies the hard constraint and minimizes the sum of the weights of the violated constraints of the soft constraint by finding a path that
Second means for obtaining an integer solution based on a sequence generation method in which the solution of the optimization problem of each stuff is a sequence and a branch and bound method;
A program for causing the computer to function as a computer. The program according to claim 1,
The first means is based on a sequence generation method that uses a sequence of solutions obtained by encoding a shift into a ZDD with a one-hot code for hard constraints and soft constraints other than cardinality constraints for each stuff, and a branch and bound method. , a program for causing the computer to function to integerize the solution of the optimization problem for each staff. The program according to claim 1,
A program for causing said first means to function said computer to solve each staff optimization problem by a plurality of processors of said computer. The program according to claim 1,
third means for encoding each of said hard constraints and said soft constraints into a CNF;
fourth means for encoding into CNF integer data of a primal solution of a linear relaxation primal problem for said sequence;
A fifth means for obtaining an integer solution by inputting the CNF obtained by the third means and the fourth means into a MaxSAT solver;
A program for causing the computer to function as a computer. The program according to claim 1,
third means for encoding each of said hard constraints and said soft constraints into a CNF;
Fifth means for obtaining an integer solution by inputting the obtained CNF into a MaxSAT solver;
A sixth means for setting the integer solution obtained by the fifth means as an initial solution of the sequence generation method;
A program for causing the computer to function as a computer.

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