JP2005339178A - Schedule generating method, schedule generating device, and computer program - Google Patents

Abstract

<P>PROBLEM TO BE SOLVED: To provide a schedule generating method, schedule generating device, and a computer program capable of easily generating a schedule of a work process wherein a plurality of sub processes comprising a plurality of work stages are selectively in parallel with each other. <P>SOLUTION: The work process includes first to n-th sub processes capable of carrying out work in parallel with each other, the first sub process has an s(i) number of work stages which are work stages P<SB>i, 1</SB>-P<SB>i, s(i)</SB>, and work is carried out by any one of the sub processes. By using a selection parameter wherein a value concerning a sub process selected for carrying out work is e(=0), and values concerning the other values are ε(=-∞), the work process is described by a max-plus algebraic value maintaining linearity. A schedule is generated by determining a selection parameter, and calculating a time (a point of time) for inputting a work object by an arithmetic equation described by the max-plus algebraic value maintaining linearity. <P>COPYRIGHT: (C)2006,JPO&NCIPI

Description

  The present invention relates to a method for creating a work process schedule for performing work through a plurality of work steps, a schedule creation device, and a computer program.

  A work system such as a production system for producing a product or a physical distribution system is composed of a plurality of work processes, and an order relationship exists between these work processes. Each work process requires a specific work time. Such a schedule for a work process composed of a plurality of work processes needs to be planned so as not to cause a delivery delay while complying with the order relationship between the work processes and the restrictions on the work time inherent in each work process. Conventionally, a method of creating a schedule by creating a Gantt chart has been used. The end time of the final work process is defined by the delivery date, and the start time of each work process is determined by calculating backward from the delivery date. The Gantt chart is created by a method such as manual creation or by a calculation method using combination optimization, and a work process schedule is managed according to the created Gantt chart.

  Another method for creating work process schedules is to use max-plus algebra. This method is disclosed in Non-Patent Document 1, for example. Express the real number field as R, D = R∪ {± ∞}, and add x (circle to +) and multiplication (circle to x) in the max-plus algebra for x and y where x and y∈D. It is defined by the following formula.

Here, max {x, y} indicates that the larger value of x and y is used as the result of the operation. The unit elements for the addition and multiplication of the max-plus algebra are −∞ and 0, respectively, according to the definition of the equation (1), and are represented by ε and e as follows.
Unit of addition: −∞ = ε
Unit of multiplication: 0 = e
Besides the expression (1), two operators ∧ and \ are also defined by the following expression.

  Here, min {x, y} indicates that the smaller value of x and y is the result of the operation. For the max-plus algebra addition and multiplication operators, multiplication has a higher priority than addition as in normal algebra. In the max-plus algebra system, the distribution law is established in the same way as in the normal algebra system, and the algebra calculation similar to that in the normal algebra system such as transformation of an expression can be performed. A similar operator is defined for a matrix. The following matrix calculation is defined for a matrix A and B of M rows and n columns and a matrix C of n rows and p columns.

Here, [X] _ij represents the i row j column component of the matrix X. With the above definition, matrix operations having linearity can be performed in the max-plus algebra. Using the max-plus algebra as described above, the order of work performed in the work process (hereinafter referred to as event counter) is an independent variable, and the time at which work is performed in the work process is described as a function of the event counter. And create a schedule.

  FIG. 7 is a schematic diagram illustrating an example of a production system that is a work process. The production system shown in the figure includes a machine M1, a machine M2, and a machine M3 as production equipment. Machines M1 and M2 are each charged with parts that are materials, and machine parts are processed. Machine M3 is charged with parts processed in machines M1 and M2, and is input from machines M1 and M2. Is further processed to deliver one part. The input and delivery of parts are repeated one after another, and parts are processed. The following variables are defined for this production system:

x ₁ (k), x ₂ (k), x ₃ (k)
: Time when the machines M1, M2, M3 start machining the k-th part u ₁ (k), u ₂ (k)
: Time when the k-th part is put into the machines M1 and M2 y (k): Time when machining of the k-th part is finished in the machine M3 d ₁ (k), d ₂ (k), d ₃ (k )
: Machining time of the k-th part in the machines M1, M2, M3 where k is an event counter and corresponds to a part number in this production system.

The production system shown in FIG. 7 has the following constraints.
The machine M1 and the machine M2 can start machining the (k + 1) th part after the kth part has been machined and the (k + 1) th part has been input.
The machine M3 can start the machining of the (k + 1) th part after finishing the machining of the kth part and receiving the (k + 1) th part from the machines M1 and M2.

From the above constraints, for example, the time x ₁ (k + 1) when the machine M1 starts machining the (k + 1) th part is the time when the machining of the kth part is finished and the (k + 1) th part is input. It will be determined at the later of the two times. Therefore, the above constraints can be described by the following algebraic expressions.

  Since the expression (4) is composed of max and +, it can be expressed using addition and multiplication of max-plus algebra. Therefore, the equation (4) can be expressed by the following equation.

By substituting and organizing the first and second expressions into the third expression of the expression (5), the expression (5) can be expressed by the following expression.

However, in the case of the production system shown in FIG. 7, x (k), A _k , B _k , u (k + 1), and C _k are expressed by the following equations.

Expression (6) is a general expression relating to the start time and end time of work in the work process. In the production system, the number of machine n, when the number of machines to be turned a part which is a material in which an external M, the number of machines that dispensing the processed parts to the outside and P, A _k is n × An n matrix, B _k is an n × M matrix, and C _k is a P × n matrix. When the same expression is calculated for the (k + 1) th and later using the expression (6) repeatedly, the following expression is obtained.

By multiplying both sides of each equation of equation (7) by C _{k + 1} , C _{k + 2} ,..., C _{k + N} from the front, the following equations can be obtained.

Here, ε _PM is a P × M matrix in which the value of each element is ε. In the production system, the time Y (k + 1) at which the processing of the future part is finished is calculated from the current internal state x (k) and the future part insertion time U (k + 1) in the production system. Is shown. Therefore, when the internal state x (k) of the production system is determined, and the desired end time R (k + 1), which is the delivery date in the production system, is defined as Y (k + 1), an appropriate part that does not delay delivery date The input time U (k + 1) can be obtained. In order to obtain an appropriate input U (k + 1), U (k + 1) satisfying the following equation obtained by substituting R (k + 1) for Y (k + 1) in equation (8) may be obtained.

  Equation (9) indicates that the larger of the first term or the second term on the right side is equal to the left side. When the first term on the right side is equal to or smaller than the left side, the second term on the right side is equal to the left side, and therefore the following equation is established from equation (9).

  If the first term on the right side of equation (9) is larger than the left side, U (k + 1) that satisfies equation (9) does not exist, and a delay in delivery is unavoidable in the current internal state. In this case, no matter what the value of U (k + 1) is, a delivery delay more than the unavoidable minimum delay occurs. In this case, it is desirable that the loading time of the component is the latest among the loading times that cause a minimum delay. The processing end time in this case is obtained on the right side of equation (9), and even if U (k + 1) is changed, the first term on the right side of equation (9) does not change, so U (k + 1) is changed. Thus, when the second term on the right side of equation (9) is made smaller than the first term on the right side of equation (9), an end time with a minimum delay is obtained. Since the second term on the right side of equation (9) increases as U (k + 1) is increased, the maximum U (k + 1) among the component insertion times U (k + 1) that cause the minimum delay is ( It is obtained when the first term on the right side of equation (9) is equal to the second term on the right side of equation (9). That is, the following equation holds for the optimum U (k + 1) in this case.

  The following formula is obtained by combining the formulas (10) and (11).

  Equation (12) is in the form of simultaneous linear equations for U (k + 1) in the max-plus algebra system. This equation can use the solution of the maximum partial decomposition in the max-plus algebra. The maximum partial decomposition is a simultaneous linear equation with respect to Z given by the following equation (13). If a solution satisfying the equation is found, the solution is given, and if a solution satisfying the equation is not found, the right side This is the solution that gives the maximum value on the left-hand side within the range not exceeding. The maximum partial decomposition of the equation (13) is given by the following equation (14).

  Therefore, the maximum partial decomposition of the equation (12) is given by the following equation.

  If the product can be completed exactly at the time of delivery with respect to the delivery time R (k + 1) determined in the production system, the equation (15) is for parts that complete the product just at the time of delivery. If the input time U (k + 1) is obtained and it is impossible to complete the product at the exact time of delivery, the completion of the product is not delayed by the delivery date and the latest input time U (k + 1) is obtained. Show. In the equation (15), the input time is obtained for all cases where the event counter is k + 1, k + 2,..., K + N, but the input time is calculated when the event counter is k + 1 using the following equation (16). Only seek.

The input time after k + 2 is obtained by recalculating while incrementing the event counter by one. By using the equation (16), it is possible to obtain an appropriate input time in the production system, that is, an appropriate work start time of the work process, and a work process schedule can be created.
Shutter (B. De Schutter), Boon (T. van den Boom) "Model predictive control for max-plus-linear discrete event systems" (UK), Auto Automatica, 2001, 37, 7 p. 1049-1056 F. Baccelli and two others, “Free-Choice Petri Nets-An Algebraic Approach” (USA), i-Triple Transactions on Automatic Control (IEEE Trans. Autom. Control), 1996, Vol. 41, No. 12, p. 1751-1777

  As described above, the max-plus algebra can well describe a work process in which a plurality of work processes are synchronized to obtain a work result. However, when a plurality of work processes are selectively arranged in parallel, it is difficult to describe the work process in the work process.

  FIG. 8 is a schematic diagram illustrating an example of a production system that is a work process in which a plurality of work processes are selectively arranged in parallel. The production system shown in the figure includes machines M1, M2, and M3 as production facilities, and the machines can work in parallel with each other. Furthermore, a selector is provided as a step before the machines M1, M2, and M3. When a material part is input to this production system, the selector selects one of the machines M1, M2, and M3, the part is input to the machine selected by the selector, and the selected machine is input. The processed parts are processed to deliver the processed parts. In FIG. 8, the machine M1 is currently selected. Other examples of work processes in which a plurality of work processes are selectively parallel include a physical distribution or communication system having paths parallel to each other, a reception system that receives customers at a plurality of reception windows, and the like.

  As an example of a method for describing a work process in which a plurality of work steps are selectively arranged in parallel, there is a method disclosed in Non-Patent Document 2 that incorporates another operation in the max-plus algebra. However, this method has a problem that the linearity that is a feature of the max-plus algebra is lost, so that the description of the work process is complicated and it is difficult to create a schedule.

  The present invention has been made in view of such circumstances, and the object of the present invention is to describe a work process with a max-plus algebra that maintains linearity, so that a plurality of work processes can be selectively performed. It is an object of the present invention to provide a schedule creation method, a schedule creation apparatus, and a computer program that can easily create a schedule of parallel work processes.

  Another object of the present invention is to provide a schedule creation method, a schedule creation device, and a computer program capable of reducing the calculation time for creating a work process schedule.

  The schedule creation method according to the first invention includes a plurality of sub-processes capable of performing work in parallel with each other, and each sub-process receives a work target input from the outside, and performs work on the received work target. The work target is input for a work process that is composed of one or a plurality of work steps arranged in the order of work performed and delivering the work result to the outside, and in which sub-process the work target can be selected. In the method of creating a work schedule in the work process by determining the sub-process selection and the work object input time by the max-plus algebra, the value is 0 for the sub-process selected as the sub-process for performing a specific work. Max-p that maintains linearity by using a selection parameter whose value is -∞ for other sub-processes The input time is calculated using a calculation formula described in the lus algebra.

The schedule creation method according to the second invention includes a first sub-process to an n-th sub-process (where n is a natural number) capable of performing work in parallel with each other, and an i-th sub-process (where i is a natural number). 1 ≦ i ≦ n) accepts work objects input from the outside, performs work on the accepted work objects, and arranges the work results in the order of work to be delivered to the outside (however, s (i ) Is a natural number) of work processes, and a work process in which a work target can be selected in which sub-process is input using a computer including a storage unit and a calculation unit. process selection and poured point of the work object determined by max-plus algebra, a method of scheduling a task in the work process, the present time t _0, the subprocess INCLUDED 1 to s (i) th time of starting the k-th task in the work process of _{{x i, j (k)} : 1 ≦ i ≦ n, 1 ≦ j ≦ s (i)} ( where, j is a natural number), work time required for work in each work process included in each sub-process {d _{i, j} : 1 ≦ i ≦ n, 1 ≦ j ≦ s (i)}, and k + 1,..., k + N A step of storing the time limit {r (k + m): 1 ≦ m ≦ N} (where m and N are natural numbers) at the time point when the work result in the th work is provided, and the k + m th work is performed. When the i-th sub-process is selected as a sub-process, δ _i (k + m) = e (where e = 0), and a sub-process other than the i-th sub-process is selected as a sub-process for performing the k + m-th work. If the _{δ i (k + m) =} ε ( where, ε = -∞) to become selected parameters {δ _i (k + m) 1 ≦ i ≦ n, 1 ≦ m ≦ N value of} a selection parameter determining step of determining by the operation unit, the matrix T ₀ of N rows one column value of each component is t _0, N given by the following formula A row-and-column matrix R (k + 1),
R (k + 1) = [r (k + 1) r (k + 1)... R (k + m)] ^T
(S (1) +... + S (n)) row and column matrix x (k) given by the following equation:
x (k) = [x _1,1 (k) x _1,2 (k) ... x _{1, s (1)} (k) x _2,1 (k) ...
x _{2, s (2)} (k) ... x _{n, 1} (k) ... x _{n, s (n)} (k)] ^T
It contains max-plus algebraic multiplication (circle x) and addition (circle +), and is given by the following equation containing a matrix A _i 'of s (i) rows s (i) columns (s (1 ) +... + S (n)) row (s (1) +... + S (n)) column matrix A _k ,

(S (1) +... + S (n)) row and column matrix B _k given by the following equation:

(S (i) -1) ε components and d _{i, s (i)} and δ _i (k) are given by the following formulas alternately including components multiplied by the multiplication of the max-plus algebra. One row (s (1) +... + S (n)) column matrix C _k ,

A matrix Γ _k given by

And a matrix Δ _k given by

calculating the time point {u (k + m): 1 ≦ m ≦ N} at which the k + 1,..., k + Nth work object is input, using the following equation including the Θ operation of the max-plus algebra.

It is characterized by including.

In the schedule creation method according to a third aspect of the present invention, in the selection parameter determination step, the calculation unit includes a penalty for delaying a scheduled time point of delivery of the work result determined by determining a value of the selection parameter from the deadline. The value of the selection parameter is determined so that the value of the evaluation function including the work cost required for the work is minimized, and in the selection parameter determination step, the evaluation function obtained from the selection parameter having a fixed value ^{N N} kinds of selection parameter values by a branch and bound method in which a combination of selection parameter values is selected by comparing a lower bound value of the parameter and a provisional minimum value that is a provisionally obtained minimum value of the evaluation function. Searching for a combination of values of selection parameters that gives the minimum value of the evaluation function from among the combinations of And wherein the Mukoto.

  According to a fourth aspect of the present invention, there is provided a schedule creation method comprising: calculating a time point u (k + 1) at which the (k + 1) -th work target is input by the arithmetic unit according to the following formula including a matrix [e ε. ,

  Calculating a matrix x (k + 1) obtained by the following equation in the calculation unit;

Storing x (k + 1) calculated by the calculation unit as new x (k) in the storage unit; a new current time point t ₀ and a new deadline {r (k + m): 1 ≦ m ≦ N } In the storage unit, and δ _i (k + m + 1) is a new δ _i (k + m) (where 1 ≦ m ≦ N−1) and δ _i (k + N) is a new δ _i (k + N). Storing the value of the evaluation function obtained from the selected parameter as the initial value of the provisional minimum value in the calculation unit, and re-creating a schedule using the value stored in the storage unit in the calculation unit The method further includes a step.

A schedule creation device according to a fifth aspect of the present invention includes first to n-th sub-processes (where n is a natural number) capable of performing work in parallel with each other, and i-th sub-process (where i is a natural number) 1 ≦ i ≦ n) accepts a work object input from outside, performs work on the accepted work object, and arranges work results in the order of work to be delivered to the outside (provided that, s (i) is a natural number), and for a work process that can select which sub-process the work target is input to, select the sub-process into which the work object is input and the time when the work object is input -plus determined by algebraic an apparatus for creating a schedule for working in the work process, k at the current time point t _0, 1 to s included in each sub-process (i) th individual work processes Time of starting the eye work _{{x i, j (k)} : 1 ≦ i ≦ n, 1 ≦ j ≦ s (i)} ( where, j is a natural number), at each working step included in each sub-process Work time required for work {d _{i, j} : 1 ≦ i ≦ n, 1 ≦ j ≦ s (i)}, and time limit when work results in the k + 1,..., K + Nth work are provided {r ( k + m): 1 ≦ m ≦ N} (where m and N are natural numbers), and δ _i (k + m) = when the i-th sub-process is selected as the sub-process performing the k + m-th operation. e (where e = 0), and δ _i (k + m) = ε (where ε = −∞) when a sub-process other than the i-th sub-process is selected as the sub-process performing the k + m-th work. selection parameters _{{δ i (k + m)} : 1 ≦ i ≦ n, 1 ≦ m ≦ n} selection parameters determine to determine the value of Means a matrix T ₀ of N rows one column value of each component is t _0, the matrix R of the N rows a row given by the following equation (k + 1),
R (k + 1) = [r (k + 1) r (k + 1)... R (k + m)] ^T
(S (1) +... + S (n)) row and column matrix x (k) given by the following equation:
x (k) = [x _1,1 (k) x _1,2 (k) ... x _{1, s (1)} (k) x _2,1 (k) ...
x _{2, s (2)} (k) ... x _{n, 1} (k) ... x _{n, s (n)} (k)] ^T
It contains max-plus algebraic multiplication (circle x) and addition (circle +), and is given by the following equation containing a matrix A _i 'of s (i) rows s (i) columns (s (1 ) +... + S (n)) row (s (1) +... + S (n)) column matrix A _k ,

(S (1) +... + S (n)) row and column matrix B _k given by the following equation:

(S (i) -1) ε components and d _{i, s (i)} and δ _i (k) are given by the following formulas alternately including components multiplied by the multiplication of the max-plus algebra. One row (s (1) +... + S (n)) column matrix C _k ,

A matrix Γ _k given by

And a matrix Δ _k given by

means for calculating the time point {u (k + m): 1 ≦ m ≦ N} at which the k + 1,..., k + Nth work object is input by the following formula including the Θ operation of the max-plus algebra.

It is characterized by providing.

In the schedule creation device according to the sixth aspect of the invention, the selection parameter determination means is configured to determine the value of the selection parameter, the penalty for delaying the scheduled delivery time of the work result from the deadline, and the work cost required for the work. The value of the selection parameter is determined so that the value of the evaluation function including the value is minimized, and is temporarily obtained from the lower bound value of the evaluation function obtained from the selection parameter having a fixed value. The evaluation function is selected from n ^N combinations of selection parameter values by a branch and bound method that selects a combination of selection parameter values by comparing with a provisional minimum value that is a minimum value of the evaluation function. Means for searching for a combination of values of the selection parameter that gives the minimum value of.

  According to a seventh aspect of the present invention, there is provided a schedule creation device that calculates a time point u (k + 1) at which the (k + 1) -th work target is input according to the following formula including a matrix [e ε.

And a start time calculating means for calculating a matrix x (k + 1) obtained by the following equation:

The storage means stores x (k + 1) calculated by the start time calculation means as new x (k), a new current time t ₀ and a new deadline {r (k + m): 1 ≦ means for storing m ≦ N}, and δ _i (k + m + 1) is a new δ _i (k + m) (where 1 ≦ m ≦ N−1), and δ _i (k + N) is a new δ _i (k + N). Means for storing the value of the evaluation function obtained from the selected parameter as an initial value of the temporary minimum value, and further comprising means for re-creating a schedule using the value stored in the storage means. Features.

A computer program according to an eighth aspect of the present invention includes a first sub-process to an n-th sub-process (where n is a natural number) capable of performing operations in parallel with each other, and an i-th sub-process (where i is a natural number). 1 ≦ i ≦ n) accepts a work object input from outside, performs work on the accepted work object, and arranges work results in the order of work to be delivered to the outside (however, s (i) Is a natural number), and for the work process that can select which sub-process the work target is to be input, the current time point t ₀ , the 1st to s (i) th included in each sub-process When the k-th work is started in each work process {x _{i, j} (k): 1 ≦ i ≦ n, 1 ≦ j ≦ s (i)} (where j is a natural number), included in each sub-process Work required for each work process During _{{d i, j: 1 ≦} i ≦ n, 1 ≦ j ≦ s (i)}, and k + 1, ..., k + N -th work result in the operation of the time it is let out deadline {r (k + m): 1 ≦ In a computer storing m ≦ N} (where m and N are natural numbers), the selection of the sub-process to which the work target is input and the time when the work target is input are determined by the max-plus algebra, and the work in the work process is performed. When the i-th subprocess is selected as the sub-process for performing the k + m-th work, δ _i (k + m) = e (provided that e = 0) and k + m-th Selection parameter {δ _i (k + m): 1 ≦ i ≦ when δ _i (k + m) = ε (where ε = −∞) is selected when a sub-process other than the i-th sub-process is selected as a sub-process to perform work. n, ≦ m ≦ N} and selecting parameter determination procedure for determining the value of a computer, N rows one row of the matrix T ₀ the value of each component is t _0, the matrix R of the N rows a row given by the following equation (k + 1) ,
R (k + 1) = [r (k + 1) r (k + 1)... R (k + m)] ^T
(S (1) +... + S (n)) row and column matrix x (k) given by the following equation:
x (k) = [x _1,1 (k) x _1,2 (k) ... x _{1, s (1)} (k) x _2,1 (k) ...
x _{2, s (2)} (k) ... x _{n, 1} (k) ... x _{n, s (n)} (k)] ^T
It contains max-plus algebraic multiplication (circle x) and addition (circle +), and is given by the following equation containing a matrix A _i 'of s (i) rows s (i) columns (s (1 ) +... + S (n)) row (s (1) +... + S (n)) column matrix A _k ,

(S (1) +... + S (n)) row and column matrix B _k given by the following equation:

(S (i) -1) ε components and d _{i, s (i)} and δ _i (k) are given by the following formulas alternately including components multiplied by the multiplication of the max-plus algebra. One row (s (1) +... + S (n)) column matrix C _k ,

A matrix Γ _k given by

And a matrix Δ _k given by

a procedure for calculating the time point {u (k + m): 1 ≦ m ≦ N} when the k + 1,..., k + Nth work object is input by the following formula including the Θ operation of the max-plus algebra;

It is characterized by including.

In the computer program according to the ninth aspect of the invention, the selection parameter determination procedure requires a penalty and work for the scheduled time point of delivery of the work result determined by determining the value of the selection parameter to the computer is later than the deadline. A value of the selected parameter so as to minimize the value of the evaluation function including the work cost, and a lower bound value of the evaluation function obtained from the selected parameter with a fixed value. Among the n ^N combinations of selected parameter values, the branch and bound method is used to select a combination of selected parameter values by comparing with a provisional minimum value that is the minimum value of the evaluation function that is provisionally obtained. To search for a combination of values of selection parameters that give the minimum value of the evaluation function.

  A computer program according to a tenth aspect of the invention is a procedure for causing a computer to calculate a time point u (k + 1) at which the (k + 1) -th work object is input according to the following formula including a matrix [e ε.

A procedure for causing a computer to calculate a matrix x (k + 1) obtained by the following equation;

A procedure for causing the computer to calculate x (k + 1) as a new x (k); a procedure for causing the computer to update the current time point t ₀ and the deadline {r (k + m): 1 ≦ m ≦ N}; The computer obtains the above-mentioned selection parameters obtained from δ _i (k + m + 1) as new δ _i (k + m) (where 1 ≦ m ≦ N−1) and δ _i (k + N) as new δ _i (k + N). The method further includes a procedure for setting the value of the evaluation function as an initial value of the provisional minimum value, and a procedure for causing the computer to create a schedule again.

  In the present invention, it is possible to describe a work process in which a plurality of work processes are selectively parallel while maintaining the linearity of the max-plus algebra. Using the production process shown in FIG. 8 as an example of the work process, a method of describing the work process using a max-plus algebra that maintains linearity will be described.

First, consider the machine M1 in the production process shown in FIG. When the machine M1 is selected as the machine for processing the (k + 1) th part, the time (time point) x ₁ (k + 1) at which the machine M1 starts processing the (k + 1) th part is kth Since it is determined at the later time of the time when the machining of the part is finished and the time u ₁ (k + 1) when the (k + 1) -th part is put in, it is expressed by the following equation.

Here, d ₁ is the machining time of the part in the machine M1. When the machine M1 is not selected as the machine for processing the (k + 1) th part, the value of the time x ₁ (k + 1) when the machine M1 starts processing the (k + 1) th part is kth The value of time x ₁ (k) at which the machining of the part is started is inherited. That is, the following formula is established.

Here, the selection parameter δ _i (k) is defined as shown in the following equation.

In the production process shown in FIG. 8, when the machine M1 is selected as the machine for processing the (k + 1) th part, δ ₁ (k) = e, δ ₂ (k) = δ ₃ (k) = ε. By integrating the equations (17) and (18) using this selection parameter, the following equation is established.

Since the value of d ₁ is equal to or greater than 0, the following equation holds for the expression in {} included in the equation (19).

  Therefore, when formula (19) is arranged, the following formula is obtained.

Machine M2 is the (k + 1) -th for the part time x ₂ to start processing of the (k + 1) and machine M3 is the (k + 1) th time x ₃ to start the machining of the component (k + 1) is also, d _2, d ₃ Assuming that the machining time is for parts in the machines M2 and M3, the following equation holds.

  Furthermore, the following equation holds for time y (k) at which the machining of the kth part ends.

By combining the expressions (20) to (23), the same expression (6) as the conventional one can be obtained. However, in the case of the production system shown in FIG. 8, x (k), A _k , B _k , and C _{k in} the equation (6) are expressed by the following equations.

Next, consider a more general work process. FIG. 9 is a schematic diagram showing another example of a production system that is a work process in which a plurality of work steps are selectively arranged in parallel. The production system shown in FIG. 9A includes three production lines that can be selected by the selector, the first line includes the machine M1, the second line includes the machine M2, and the third line. Comprises a machine M31 and a machine M32 in series. When a material part is input to this production system, the selector selects one of the three lines, the part is input to the line selected by the selector, and the part is processed. When the third line is selected, the machine M31 first processes the part, and further the part is processed by the machine M32 to deliver the processed part. Machine M1 is (k + 1) -th time x ₁ to start the machining of the component (k + 1) and machine M2 is (k + 1) -th time x ₂ to start processing of the component (k + 1), respectively (20) and ( 21) is obtained by the equation. The time x ₃₁ (k + 1) at which the machine M31 starts machining the (k + 1) -th part is expressed by the following expression, assuming that d ₃₁ is the machining time of the part in the machine M31 from the expression (22).

The time x ₃₂ (k + 1) when the machine M32 starts machining the (k + 1) th part is the time when the machine M32 finishes machining the kth part and the machine M31 finishes machining the (k + 1) th part. Since it is determined at the later time of the set times, it is expressed by the following formula.

  Here, the following equation is defined for a in which aεD.

Further, since d ₃₁ > 0, the following formula is established.

  Therefore, by substituting (24) into (25) and rearranging, (25) becomes the following:

  Further, the following equation holds for time y (k) at which the machining of the k-th component ends.

By combining equations (21), (22), (24), (26), and (27), equation (6) is obtained. However, in the case of the production system shown in FIG. 9A, x (k), A _k , B _k , and C _k in equation (6) are expressed by the following equations.

The production system shown in FIG. 9B is a more general production system, and the third line includes s machines M31 to M3s in series. This production system can be similarly described by the equation (6). The time at which the machine M3j starts machining the (k + 1) -th part is x _{3, j} (k + 1), and the machining time of the part at the machine M3j is d _{3, j} . In this case, x in the expression (6) (K), A _k , B _k , and C _k are represented by the following equations.

Here, the matrix A _k is a matrix of (s + 2) × (s + 2). The matrix A ′ ₃ included in the matrix A _k is an s × s matrix, and corresponds to a lower triangular matrix in a normal algebra. Further, ε _{1, s} and ε _{s, 1} included in the matrix A _k are 1 × s and s × 1 matrices, respectively, in which the value of each element is ε. The matrices x (k) and B _k are (s + 2) × 1 matrices, and the matrix C _k is a 1 × (s + 2) matrix.

FIG. 10 is a schematic diagram showing a general work process in which a plurality of work processes are selectively arranged in parallel. The work process includes n sub-processes of a first sub-process to an n-th sub-process capable of performing work in parallel with each other, and the i-th sub-process includes work steps P _{i, 1 to} P _{i, s ( i)} s (i) work steps. A selector is provided in front of the work flow of each sub-process, and when performing the work, the selector selects one of the sub-processes, and the work is performed in the selected sub-process. Each work process P _{i, j} of the i-th sub-process receives work objects from other work processes P _{i, j-1 at} all stages, performs work on the work objects, and obtains work results as other work in the subsequent stage. They are arranged in order of work so as to be input to the process P _{i, j + 1} . Incidentally working process P _{i, 1} to P i including the i-th sub-process, among _{s (i),} the working process P _{i, 1} of the first stage are charged work object from the selector, the final stage working process P _{i, s (i)} provides the final work result of the i-th sub-process. This work process can be similarly described by the equation (6). The selection parameter δ _i (k) is δ _i (k) = e when the k-th operation is performed in the i-th sub-process, and δ when the k-th operation is performed in a sub-process other than the i-th sub-process. _i (k) = ε. In this case, x (k), A _k , B _k , C _{k in} the equation (6) is expressed by the following equation.

Here, x _{i, j} (k) is the time when the work process P _{i, j} starts the k-th work, and d _{i, j} is the work time in the work process P _{i, j} . The matrix A _k is a matrix of (s (1) +... + S (n)) × (s (1) +. A matrix A ′ _i included in the matrix A _k is a matrix of s (i) × s (i), and corresponds to a lower triangular matrix in a normal algebra. Further, ε _{a, b} included in the matrix A _k is an a × b matrix in which the value of each element is ε. The matrices x (k) and B _k are (s (1) +... + S (n)) × 1 matrix, and the matrix C _k is a 1 × (s (1) +... + S (n)) matrix. is there.

As described above, by introducing a selection parameter, a work process in which a plurality of work steps are selectively paralleled can be described by Equation (6) using a max-plus algebra that maintains the same linearity as in the past. It becomes possible. Therefore, by using the equations (15) and (16) as in the prior art, it is possible to obtain an appropriate work object input time u (k + 1) in a work process in which a plurality of work steps are selectively parallel. It is possible to create a work process schedule. However, Δ _k in this case is expressed by the following equation.

In the first, second, fifth, and eighth inventions, a plurality of work steps are selectively represented by expressing x (k), A _k , B _k , and C _k in equation (6) using selection parameters. The calculation formula that calculates the input time (time) of the work target in the work process that is parallel to is described in the max-plus algebra that maintains linearity, and the work target is input using the calculation formula that determines the selection parameter. Calculate the time.

  In the third, sixth, and ninth inventions, the branch and bound method is used to combine the optimum selection parameter values that minimize the value of the evaluation function including the delivery delay penalty and the work cost. To explore.

In the fourth, seventh and tenth inventions, when calculating the input time u (k + 1) of the work object in the event counter k + 1 and creating a schedule in which k is counted up, 1 ≦ m ≦ N−1 In this range, δ _i (k + m + 1) is set as a new δ _i (k + m), and δ _i (k + N) is set as a new δ _i (k + N). Use as the initial minimum value.

In the first, second, fifth and eighth inventions, x (k) in the formula (6) using the selection parameter {δ _i (k + m): 1 ≦ i ≦ n, 1 ≦ m ≦ N}. , A _k , B _k , and C _k , a work process in which a plurality of work processes are selectively parallel can be described by a max-plus algebra that maintains linearity. Therefore, by determining the value of the selection parameter and substituting the value of the selection parameter into the formula described in the max-plus algebra that maintains linearity, the work process is calculated by calculating the input time (time) of the work target. Can be easily created.

  In the third, sixth, and ninth inventions, until the evaluation function is calculated by using a branch and bound method that searches for an optimal combination of selection parameter values that minimizes the evaluation function value In addition, since the optimum selection parameter is searched while excluding the combination of the selection parameters that are obviously not optimum, the calculation time for searching for the optimum selection parameter can be shortened.

In the fourth, seventh and tenth inventions, the previously obtained optimal value δ _i (k + m + 1) is set as a new δ _i (k + m), and the previous optimal value δ _i (k + N) is set as a new value. Since the selection parameter set as δ _i (k + N) is likely to be a value close to the optimal selection parameter in the work process in which k is counted up, the initial value of the provisional minimum value of the evaluation function is the minimum value of the evaluation function. The present invention has an excellent effect that the value is very close to the value and the calculation time for searching for the optimum selection parameter by the branch and bound method is further shortened.

Hereinafter, the present invention will be specifically described with reference to the drawings showing embodiments thereof.
FIG. 1 is a block diagram showing the internal configuration of the schedule creation device 1 of the present invention. The schedule creation device 1 is configured using a general-purpose computer such as a personal computer or a server device. The schedule creation device 1 includes a CPU (arithmetic unit) 11 that performs computation, a RAM (storage unit) 12 that stores temporary information generated along with the computation, an external storage device 13 such as a CD-ROM drive, And an internal storage device 14 such as a hard disk. The CPU 11 reads the computer program 20 of the present invention from the recording medium 2 such as a CD-ROM by the external storage device 13 and stores the read computer program 20 in the internal storage device 14. The computer program 20 is loaded from the internal storage device 14 to the RAM 12 as necessary, and the CPU 11 executes processing necessary for the schedule creation device 1 based on the loaded computer program 20.

  The schedule creation device 1 includes an input device 15 including a device for inputting information by an operator's operation such as a keyboard or a device for inputting information necessary for schedule creation from an external device (not shown). The schedule device 1 includes an output device 16 including a display device such as a CRT display, a printer device, or a device that outputs a created schedule to an external device (not shown).

  The computer program 20 of the present invention may be loaded from an external server device (not shown) connected to the schedule creation device 1 to the schedule creation device 1 and stored in the internal storage device 14.

The purpose of the processing performed by the schedule creation device 1 of the present invention is to create an optimal schedule for the work process. When the value of the selection parameter δ _i (k + m) is determined, the formulas (15) and (16) are used. Since the input time (time point) u (k + 1) of the work target can be obtained, it is necessary to obtain an optimum value of δ _i (k + m) in order to obtain an optimum schedule. Let n be the number of sub-processes included in the production process, and N be the number of steps indicating the number of steps ahead including the event counter when u (k + 1) is calculated. At this time, δ _i (k + m) ε {ε, e}, 1 ≦ i ≦ n, and 1 ≦ m ≦ N. Further, when the value of m is fixed to m = N ₀ , δ _i (k + N ₀ ) can have a value that any one of n δ _i (k + N ₀ ) is e, The value is ε. Therefore, when the value of m is fixed at m = N _0, there are n combinations of values that δ _i (k + N ₀ ) can take. That is, there are n ^N combinations of values that the selection parameter δ _i (k + m) can take.

In the present invention, a branch and bound method is used as a method of selecting an optimal combination from among n ^N combinations of δ _i (k + m). In the branch and bound method, an evaluation function for evaluating a combination of δ _i (k + m) is defined, and an optimum one of n ^N types of δ _i (k + m) combinations based on the value of the evaluation function is defined. This is a method of searching for combinations. For example, the value of δ _i (k + m) from m = 1 to N ₀ is provisionally determined, and the range of evaluation functions that can be taken in this case is obtained. When the range of the obtained evaluation function does not include a value that is better than the optimum provisional value of the evaluation function, the optimum value cannot be obtained regardless of the combination of the values of δ _i (k + m) after m = N ₀ +1. Therefore, the further evaluation of the combination of δ _i (k + m) including the provisionally determined value combination is stopped, and the other combinations are evaluated. When the range of the obtained evaluation function includes a value that is better than the optimal provisional value of the evaluation function, there is a possibility that an optimum combination exists in the combination of δ _i (k + m) including the combination of provisionally determined values. There is. Accordingly, the value of δ _i (k + m) of m = N ₀ +1 is further determined, and the range of the evaluation function that can be similarly obtained is obtained. When the above processing is repeated and the value of δ _i (k + m) for m = 1 to N is provisionally determined, the evaluation function is calculated, and the calculated evaluation function value is better than the provisional optimum value. Sometimes, the temporary optimal value is updated, and the combination of δ _i (k + m) in this case is set as the temporary optimal combination. In the branch and bound method, as described above, the combinations of δ _i (k + m) are evaluated while excluding combinations that do not need to be evaluated, and the combination of δ _i (k + m) that gives the optimal evaluation function value is optimized. Use a combination.

  In the present invention, J defined by the following equation is used as the evaluation function of the branch and bound method.

Here, both α and β are positive constants. R (k + m) corresponds to the delivery date of the work at the event counter k + m. Further, y (k + m) is a value of an element included in Y (k + 1) calculated by the equation (8) using U (k + 1) obtained by the equation (15), and the work is scheduled to end at the event counter k + m. It's time. Therefore, the second term of the equation (30) indicates a penalty when a delivery delay occurs. Further, the first term of the expression (30) indicates the work cost when performing the work in the work process. In the present invention, it is assumed that the work cost is lower as the sub-process takes a longer time for the work. Therefore, the value of the first term of the equation (30) becomes smaller as the working time in the sub-process selected to perform the work is longer. When searching for the optimum combination of δ _i (k + m) in the present invention, the combination of δ _i (k + m) that gives the minimum value of J defined by the equation (30) is searched.

  FIG. 2 is a flowchart showing a procedure of processing performed by the schedule creation device 1 of the present invention. The schedule creation device 1 performs processing for obtaining an input time u (k + 1) of an appropriate work target for the work process as shown in FIG. The CPU 11 of the schedule creation device 1 executes the following processing according to the computer program 20 of the present invention loaded on the RAM 12.

First, the CPU 11 of the schedule creation device 1 sets initial parameters necessary for schedule creation processing (S1). At this time, the CPU 11 determines the work start time {x _{i, j} (k): 1 ≦ i ≦ n, 1 ≦ j ≦ s (i)} for each work process P _{i, j at} the current time t ₀ and the event counter k. the a component (28) state variables x (k) defined by the equation, each work process P _i, the system parameter is a working time at _{j {d i, j: 1} ≦ i ≦ n, 1 ≦ j ≦ s (i)}, number of steps N, latest delivery date {r (k + m): 1 ≦ m ≦ N}, which is the deadline of the work end time in event counter k + m, initial value of provisional minimum value J _min of evaluation function J Set. The CPU 11 accepts these values from the outside by the input device 15 or sets the initial parameters by reading the values stored in the internal storage device 14 in advance into the RAM 12. When setting the initial value of the provisional minimum value _Jmin , an evaluation function obtained by setting the values of the selection parameters {δ _i (k + m): 1 ≦ i ≦ n, 1 ≦ m ≦ N} to predetermined values. The value of J may be set as the initial value of the provisional minimum value _Jmin . Further, the schedule creation device 1 may be provided with means for measuring the current time t ₀ .

Next, the CPU 11 performs a selection parameter optimization process for obtaining an optimal combination of the selection parameters δ _i (k + m) (S2). 3 and 4 are flowcharts showing the processing procedure of the subroutine of the selection parameter optimization processing in step S2. The CPU 11 first sets m = 1 (S201).

Next, the CPU 11 sets each unique value of δ _i (k + m) to e by setting the unique value of {δ _i (k + m): 1 ≦ i ≦ n} in which m is determined to be e. Is fixed (S202). Next, the CPU 11 determines whether or not m = N (S203). If the value of m was smaller than N (S203: NO), CPU11 is the current m as N _0, selection parameters including a combination of fixed values with m = 1~N ₀ δ _i ( The lower bound value J ′, which is the lower limit value of the evaluation function J obtained by the combination of k + m), is calculated (S204).

Here, a method of calculating the lower bound value J ′ will be described. First, consider the first term of equation (30). D _i included in the first term of the equation (30) is the total of the work times of s (i) work steps included in the i-th sub-process. When the maximum value among the n d _i and d _M, since it is d _i> 0, the definition of _{j 1 (k + m),} j 1 (k + m) ≧ -d M holds. This indicates that the work cost is lowest when a sub-process that requires the most work is selected. Therefore, the following equation holds for the first term of equation (30).

  Next, consider the second term of equation (30). Equation (8) indicates that the larger value of the first term and the second term on the right side is equal to the value on the left side. Therefore, the following formula is established from the formula (8).

Thereby, the following formula is further established by decomposing Y (k + 1) and Γ _k into respective elements.

J ₂ '(k + m) is defined by the following equation.

(32), (33), and the definition of (30) j ₂ in the second term of the equation _{(k + m), j 2} (k + m) ≧ j 2 '(k + m) is established. Therefore, the following equation holds for the second term of equation (30).

Therefore, the lower bound value J ′ of the evaluation function J obtained by the combination of the selection parameters δ _i (k + m) including the combination of values fixed at m = 1 to N ₀ is defined by the following equation.

  From Equations (31) and (34), J ≧ J ′, and the value of the evaluation function J is always greater than or equal to the lower bound value J ′. In step S204, the CPU 11 calculates a lower bound value J ′ defined by the equation (35).

Next, the CPU 11 determines whether or not the calculated lower bound value J ′ is smaller than the current provisional minimum value J _min of the evaluation function J (S205). When the value of the lower bound value J ′ is smaller than the value of the provisional minimum value J _min (S205: YES), among the combinations of selection parameters δ _i (k + m) including the combination of values fixed at m = 1 to N ₀ Since there may be a combination that gives a value of the evaluation function J smaller than the provisional minimum value J _min , the CPU 11 increments the value of m to search for a better combination including a fixed value combination. (S206), the process is returned to step S202, and the value of δ _i (k + m) is further fixed.

When the value of the lower bound value J ′ is equal to or larger than the value of the provisional minimum value J _min , the combination of the selection parameters δ _i (k + m) including the combination of values fixed at m = 1 to N ₀ is the provisional minimum value. Since there is no combination that gives a value of the evaluation function J smaller than J _min, a combination including a fixed value combination is excluded from the search range of the optimum combination.

If the value of the lower bound value J ′ is greater than or equal to the provisional minimum value J _min in step S205 (S205: NO), the CPU 11 determines the only value of {δ _i (k + m): 1 ≦ i ≦ n}. It is determined whether or not all combinations of n δ _i (k + m), where e is set to e and all other values are ε, have been tried (S207). When there is a combination that has not been tried yet among the n combinations of δ _i (k + m) (S207: NO), the CPU 11 uses another combination that has not yet tried the value of each δ _i (k + m). The process is fixed (S208), and the process returns to step S203.

If m = N in step S203 (S203: YES), the values of the selection parameters {δ _i (k + m): 1 ≦ i ≦ n, 1 ≦ m ≦ N} are all fixed. The CPU 11 calculates an evaluation function J defined by equation (30) (S211). Next, the CPU 11 determines whether or not the calculated value of the evaluation function J is smaller than the temporary minimum value J _min (S212). If the value of the evaluation function J is greater than or equal to the provisional minimum value J _min (S212: NO), the combination of the current selection parameter δ _i (k + m) is not the optimal combination, so a new selection parameter combination is selected. In order to search, the process proceeds to step S207.

Step If the value of the evaluation function J is smaller than the value of the temporary minimum value J _min in S212 (S212: YES), CPU 11 has the largest portion degradation (28) and (29) using equation (15) The input U (k + 1) based on is calculated by the following equation using an N × 1 matrix T _{0 in} which the value of each component is t ₀ (S213).

U (k + 1) is the input time of the work object in the event counters k + 1 to k + N, and the work object input time cannot be made earlier than the current time t _0, so the equation (36) holds.

Next, the CPU 11 calculates the scheduled work end time Y (k + 1) by substituting U (k + 1) into the equation (8) using the equations (28) and (29) (S214). Next, the CPU 11 determines whether or not the combination of the current selection parameter δ _i (k + m) is an executable solution (S215). At this time, the CPU 11 determines whether or not the following relationship holds for the calculated U (k + 1) and Y (k + 1).
u (k + m + 1) ≧ u (k + m), 1 ≦ m ≦ N
y (k + m + 1) ≧ y (k + m), 1 ≦ m ≦ N (37)

When these relationships do not hold, the work target input time or work end time in each event counter is not arranged in the order of the event counter. In this case, the combination of the selection parameters δ _i (k + m) is not executed. It is a possible solution. When the combination of the current selection parameter δ _i (k + m) is an infeasible solution in step S215 (S215: NO), the CPU 11 determines that the combination of the current selection parameter δ _i (k + m) is not an optimal combination. In order to search for a new combination of selection parameters, the process proceeds to step S207.

If the combination of the current selection parameter δ _i (k + m) is an executable solution in step S215 (S215: YES), the CPU 11 sets the value of the evaluation function J as the value of the new temporary minimum value _Jmin. The value J _min is updated (S216). Next, the CPU 11 stores the combination of the current selection parameter δ _i (k + m) and U (k + 1) as a provisional solution in the RAM 12 (S217), and the process proceeds to step S207 to search for a new combination of selection parameters. Proceed.

When all the combinations of n δ _i (k + m) are tried in step S207 (S207: YES), the CPU 11 determines whether m = 1 (S209). When the value of m is larger than 1 (S209: NO), the CPU 11 decrements the value of m by 1 to reduce the selection parameter δ _i (k + m) whose value is fixed and evaluate other combinations. (S210), the process returns to step S207. If m = 1 (S209: YES), the subroutine of the selection parameter optimization process in step S2 is terminated, and the process returns to the main routine. The combination of the selection parameters δ _i (k + m) and U (k + 1) stored in the RAM 12 corresponding to the provisional minimum value J _min when the selection parameter optimization processing subroutine is completed is the minimum value of the evaluation function J. The optimal solution to give.

After the selection parameter optimization process subroutine in step S2 is completed, the CPU 11 calculates the input time u (k + 1) of the work target in the event counter k + 1 according to the equation (16) substituted with the optimal solution U (k + 1). (S3) The optimal solution of the selected parameter δ _i (k + 1) in the event counter k + 1 and the calculated u (k + 1) are output from the output unit (S4). Work is performed in the work process based on the selection parameter δ _i (k + 1) and work input time u (k + 1) thus output. Next, the CPU 11 determines whether or not to end the schedule creation process (S5). If the schedule creation process is to be terminated (S5: YES), the CPU 11 ends the process. If the schedule creation process is not terminated (S5: NO), the CPU 11 updates the parameter for creating the parameters necessary for the schedule creation process at the next step in order to perform the process at the next step after incrementing the event counter. Processing is performed (S6).

FIG. 5 is a flowchart showing the processing procedure of the subroutine for parameter update processing in step S6. The CPU 11 calculates the state variable x (k + 1) in the event counter k + 1 according to the first expression (7) using u (k + 1) calculated in step S3 (S601). Next, the CPU 11 sets the latest state variable x (k) by setting the calculated value of x (k + 1) as the value of the latest state variable x (k) to count up the event counter ( S602). Next, the CPU 11 updates the value of the due date {r (k + m): 1 ≦ m ≦ N} that is the deadline of the work end time at the current time t ₀ and the event counter k + m (S603). At this time, the CPU 11 may update the value of each r (k + m) by setting r (k + m + 1) to r (k + m). If there is a change in the delivery date, the delivery date accepted by the input device 15 Is set as the latest value of r (k + m).

Next, the CPU 11 uses the combination of the selection parameters δ _i (k + m) stored in the RAM 12 as the optimum solution, and sets the value of δ _i (k + m + 1) in the optimum solution as a new value of δ _i (k + m). Then, a combination of selection parameters {δ _i (k + m): 1 ≦ i ≦ n, 1 ≦ m ≦ N} in which the value of δ _i (k + N) in the optimal solution is set as a new value of δ _i (k + N) is created. (S604). Next, the CPU 11 calculates U (k + 1) according to the expression (36) and Y (k + 1) according to the expression (8) using the created combination of selection parameters (S605). Next, the CPU 11 determines whether or not the combination of the selected parameters is an executable solution by determining whether or not the equation (37) holds for the calculated U (k + 1) and Y (k + 1) ( S606).

When the combination of the selected parameters is an executable solution in step S606 (S606: YES), the CPU 11 calculates the evaluation function J using equation (30) (S607). Next, the CPU 11 sets the initial value of the temporary minimum value J _min by setting the calculated value of the evaluation function J as the value of the new temporary minimum value J _min (S608). Next, the CPU 11 stores the combination of the currently selected parameter δ _i (k + m) and U (k + 1) in the RAM 12 as a provisional solution (S609), ends the parameter update processing subroutine in step S6, and performs the processing as the main routine. Return to. If the combination of the selection parameter is not a feasible solution in step S606 (S606: NO), CPU11 sets the initial value of the temporary minimum value J _min to a predetermined value (S610), the parameter updating process in step S6 The subroutine ends and the process returns to the main routine.

  After the parameter update process subroutine in step S6 is completed, the CPU 11 returns the process to step S2 and performs a process for creating a new schedule using the updated parameters.

As described above in detail, in the present invention, x (k), A _k , B _k , and C _k are represented using selection parameters {δ _i (k + m): 1 ≦ i ≦ n, 1 ≦ m ≦ N}. Thus, a work process in which a plurality of work processes are selectively arranged in parallel can be described by a max-plus algebra that maintains linearity. Accordingly, by determining the value of the selection parameter and substituting the determined selection parameter value into the equations (36) and (16) to calculate the input time u (k + 1) of the work object, the work process schedule is determined. Can be easily created.

  In the present invention, when determining the optimum value of the selection parameter, the optimum value is searched using the branch and bound method. Based on the lower bound value J ′ of the evaluation function J obtained by fixing some values of the selection parameters, the optimal selection is performed while excluding combinations of selection parameters that are obviously not optimal without calculating the evaluation function J. Since the parameter is searched, the calculation time for searching for the optimum selected parameter can be shortened. In addition, since the calculation amount necessary for calculating the lower bound value J ′ according to the equation (35) is significantly smaller than the calculation amount for calculating the evaluation function J, the calculation time for searching for the optimum selection parameter Is shortened more.

Further, in the present invention, when u (k + 1) obtained by counting up the event counter is calculated, δ _i (k + m + 1) is replaced with a new δ _i (k + m) using the optimum value of the selection parameter obtained previously. and then, utilizing the value of the evaluation function obtained [delta] _i and (k + N) as a new δ _i (k + N), as the initial value of the temporary minimum value J _min of the evaluation function J used in the branch and bound method. There is a high possibility that the new selection parameter that reuses the optimum value is close to the optimum selection parameter in the work process in which the event counter is counted up. Therefore, there is a high possibility that the initial value of the provisional minimum value J _min is very close to the minimum value of the evaluation function J, and the calculation time for searching for the optimum selection parameter by the branch and bound method is further shortened.

  FIG. 6 is a chart showing an example of calculation time when the schedule creation processing is performed using the present invention. For the production system shown in FIG. 9 (a), a schedule creation process was performed 60 times while counting up an event counter using a predetermined parameter value. In this chart, when the value of N is set to 1-7, all combinations of selected parameters are searched exhaustively without using the branch and bound method, and the optimum value of the selected parameter is determined. The calculation time is compared between the case where the selection parameter is used and the optimum value of the selection parameter is used when the event counter is further counted up. The calculation time indicated by a number in the chart indicates an average of the calculation time obtained by performing a plurality of processes in each case as a relative value. As the value of N increases, the calculation time when the branch and bound method is used is shorter than when the selection is interrupted, and the calculation time when the optimum value of the selected parameter is used at the time of counting up is further shortened. Yes. By increasing the value of N, it is possible to pre-read the schedule based on the delivery date to the farther future and create a schedule with a smaller delivery date delay. Therefore, it is clear that the calculation time can be shortened by using the present invention when creating a better schedule.

It is a block diagram which shows the internal structure of the schedule preparation apparatus of this invention. It is a flowchart which shows the procedure of the process which the schedule preparation apparatus of this invention performs. It is a flowchart which shows the process sequence of the subroutine of the selection parameter optimization process of step S2. It is a flowchart which shows the process sequence of the subroutine of the selection parameter optimization process of step S2. It is a flowchart which shows the process sequence of the subroutine of the parameter update process of step S6. It is a graph which shows the example of the calculation time which performed the process of schedule creation using this invention. It is a schematic diagram which shows the example of the production system which is a work process. It is a mimetic diagram showing an example of a production system which is a work process in which a plurality of work processes are selectively performed in parallel. It is a schematic diagram which shows the other example of the production system which is a work process with which several work processes are selectively parallel. It is a schematic diagram which shows the general work process in which the some work process is selectively parallel.

Explanation of symbols

1 Schedule creation device 11 CPU (calculation unit)
12 RAM (storage unit)
2 Recording medium 20 Computer program

Claims (10)

Work that includes multiple sub-processes that can work in parallel with each other, each sub-process accepts work objects input from the outside, performs work on the accepted work objects, and delivers work results to the outside For a work process that consists of one or a plurality of work steps arranged in sequence, and can select which sub-process the work target is input to, the selection of the sub-process into which the work target is input and the time when the work target is input In the method of creating a work schedule in the work process, by defining the max-plus algebra,
A calculation described by a max-plus algebra that maintains linearity by using a selection parameter with a value of 0 for the subprocess selected as the subprocess to perform a specific task and a value of -∞ for the other subprocesses. A schedule creation method characterized by calculating the input time using an equation. The first sub-process to the n-th sub-process (where n is a natural number) capable of performing work in parallel with each other, where the i-th sub-process (where i is a natural number and 1 ≦ i ≦ n) is external Is composed of s (i) (where s (i) is a natural number) work steps arranged in the order of work for receiving work targets input from the target, performing work on the received work targets, and delivering work results to the outside. For a work process that can select which sub-process the work target is input to, using a computer having a storage unit and a calculation unit, the selection of the sub-process into which the work target is input and the time when the work target is input In a method of creating a work schedule in the work process, defined by a max-plus algebra,
The current time point t ₀ , the time point at which the k-th work is started in each of the 1-s (i) -th work steps included in each sub-process {x _{i, j} (k): 1 ≦ i ≦ n, 1 ≦ j ≦ s (i)} (where j is a natural number), work time required for work in each work process included in each sub-process {d _{i, j} : 1 ≦ i ≦ n, 1 ≦ j ≦ s (i )}, And the time limit {r (k + m): 1 ≦ m ≦ N} (where m and N are natural numbers) when the work results in the k + 1,..., K + Nth work are provided are stored in the storage unit. Steps,
When the i-th sub-process is selected as the sub-process for performing the k + m-th work, δ _i (k + m) = e (where e = 0), and the sub-process for performing the k + m-th work is other than the i-th sub-process. When a sub-process is selected, the values of selection parameters {δ _i (k + m): 1 ≦ i ≦ n, 1 ≦ m ≦ N} satisfying δ _i (k + m) = ε (where ε = −∞) A selection parameter determination step determined by the calculation unit;
An N-row, one-column matrix T ₀ in which the value of each component is t ₀ , an N-row, one-column matrix R (k + 1) given by the following equation:
R (k + 1) = [r (k + 1) r (k + 1)... R (k + m)] ^T
(S (1) +... + S (n)) row and column matrix x (k) given by the following equation:
x (k) = [x _1,1 (k) x _1,2 (k) ... x _{1, s (1)} (k) x _2,1 (k) ...
x _{2, s (2)} (k) ... x _{n, 1} (k) ... x _{n, s (n)} (k)] ^T
It contains max-plus algebraic multiplication (circle x) and addition (circle +), and is given by the following equation containing a matrix A _i 'of s (i) rows s (i) columns (s (1 ) +... + S (n)) row (s (1) +... + S (n)) column matrix A _k ,

(S (1) +... + S (n)) row and column matrix B _k given by the following equation:

(S (i) -1) ε components and d _{i, s (i)} and δ _i (k) are given by the following formulas alternately including components multiplied by the multiplication of the max-plus algebra. One row (s (1) +... + S (n)) column matrix C _k ,

A matrix Γ _k given by

And a matrix Δ _k given by

calculating the time point {u (k + m): 1 ≦ m ≦ N} at which the k + 1,..., k + Nth work object is input, using the following equation including the Θ operation of the max-plus algebra.

The schedule creation method characterized by including this. In the selection parameter determination step, the calculation unit includes a penalty for delaying the scheduled time of delivery of the work result determined by determining the value of the selection parameter and the work cost required for the work. Determine the value of the selected parameter so that the value of the evaluation function is minimized,
In the selection parameter determination step, a selection is made by comparing a lower bound value of the evaluation function obtained from a selection parameter with a fixed value and a provisional minimum value which is a provisionally obtained minimum value of the evaluation function. Searching the arithmetic unit for a combination of selected parameter values that gives the minimum value of the evaluation function from n ^N selected parameter value combinations by a branch and bound method for selecting a combination of parameter values. The schedule creation method according to claim 2, further comprising: calculating the time point u (k + 1) at which the k + 1th work target is input by the calculation unit according to the following expression including a matrix [e ε ... ε] of one row and N columns;

Calculating a matrix x (k + 1) obtained by the following equation in the calculation unit;

Storing x (k + 1) calculated by the arithmetic unit in the storage unit as new x (k);
Storing the new current time point t ₀ and the new deadline {r (k + m): 1 ≦ m ≦ N} in the storage unit;
δ _i (k + m + 1) is a new δ _i (k + m) (where 1 ≦ m ≦ N−1) and δ _i (k + N) is a new δ _i (k + N). Storing a value in the calculation unit as an initial value of the provisional minimum value;
The schedule creation method according to claim 3, further comprising: creating a schedule again by the calculation unit using the value stored in the storage unit. The first sub-process to the n-th sub-process (where n is a natural number) capable of performing work in parallel with each other, where the i-th sub-process (where i is a natural number and 1 ≦ i ≦ n) is external Is composed of s (i) work steps (where s (i) is a natural number) arranged in the order of the work in which the work object input from the work object is received, the work on the received work object is performed, and the work result is delivered to the outside. For a work process in which it is possible to select which sub-process the work object is to be input, the sub-process to which the work object is input and the time point at which the work object is input are determined by the max-plus algebra, and the work process In the device for creating work schedules at
The current time point t ₀ , the time point at which the k-th work is started in each of the 1-s (i) -th work steps included in each sub-process {x _{i, j} (k): 1 ≦ i ≦ n, 1 ≦ j ≦ s (i)} (where j is a natural number), work time required for work in each work process included in each sub-process {d _{i, j} : 1 ≦ i ≦ n, 1 ≦ j ≦ s (i )}, And storage means for storing the time limit {r (k + m): 1 ≦ m ≦ N} (where m and N are natural numbers) when the work result in the k + 1,..., K + Nth work is provided,
When the i-th sub-process is selected as the sub-process for performing the k + m-th work, δ _i (k + m) = e (where e = 0), and the sub-process for performing the k + m-th work is other than the i-th sub-process. When the sub-process is selected, the value of the selection parameter {δ _i (k + m): 1 ≦ i ≦ n, 1 ≦ m ≦ N} that satisfies δ _i (k + m) = ε (where ε = −∞) is determined. Selection parameter determination means to
An N-row, one-column matrix T ₀ in which the value of each component is t ₀ , an N-row, one-column matrix R (k + 1) given by the following equation:
R (k + 1) = [r (k + 1) r (k + 1)... R (k + m)] ^T
(S (1) +... + S (n)) row and column matrix x (k) given by the following equation:
x (k) = [x _1,1 (k) x _1,2 (k) ... x _{1, s (1)} (k) x _2,1 (k) ...
x _{2, s (2)} (k) ... x _{n, 1} (k) ... x _{n, s (n)} (k)] ^T
It contains max-plus algebraic multiplication (circle x) and addition (circle +), and is given by the following equation containing a matrix A _i 'of s (i) rows s (i) columns (s (1 ) +... + S (n)) row (s (1) +... + S (n)) column matrix A _k ,

(S (1) +... + S (n)) row and column matrix B _k given by the following equation:

(S (i) -1) ε components and d _{i, s (i)} and δ _i (k) are given by the following formulas alternately including components multiplied by the multiplication of the max-plus algebra. One row (s (1) +... + S (n)) column matrix C _k ,

A matrix Γ _k given by

And a matrix Δ _k given by

means for calculating the time point {u (k + m): 1 ≦ m ≦ N} at which the k + 1,..., k + Nth work object is input by the following formula including the Θ operation of the max-plus algebra.

A schedule creation device comprising: The selection parameter determining means includes
The selection parameter is such that the value of the evaluation function including the penalty for delaying the scheduled delivery time of the work result determined by determining the value of the selection parameter and the work cost required for the work is minimized. To determine the value of
A combination of selection parameter values is selected by comparing the lower bound value of the evaluation function obtained from a selection parameter with a fixed value and the provisional minimum value, which is the provisional minimum value of the evaluation function. 6. The method according to claim 5, further comprising means for searching for a combination of values of the selection parameter that gives the minimum value of the evaluation function from among n ^N combinations of values of the selection parameter by the branch and bound method. Schedule creation device. means for calculating a time point u (k + 1) at which the (k + 1) th work object is input by the following equation including a matrix [e ε ... ε] of one row and N columns;

And a start time calculating means for calculating a matrix x (k + 1) obtained by the following equation:

The storage means
Means for storing x (k + 1) calculated by the start time calculation means as new x (k);
Means for storing the new current time t ₀ and the new deadline {r (k + m): 1 ≦ m ≦ N};
δ _i (k + m + 1) is a new δ _i (k + m) (where 1 ≦ m ≦ N−1) and δ _i (k + N) is a new δ _i (k + N). Means for storing a value as an initial value of the provisional minimum value,
The schedule creation apparatus according to claim 6, further comprising means for creating a schedule again using a value stored in the storage means. The first sub-process to the n-th sub-process (where n is a natural number) capable of performing work in parallel with each other include the i-th sub-process (where i is a natural number and 1 ≦ i ≦ n) Is composed of s (i) (where s (i) is a natural number) work steps arranged in the order of work for receiving work targets input from the target, performing work on the received work targets, and delivering work results to the outside. For a work process in which it is possible to select which sub-process the work target is to be input, the k-th work in each of the first to s (i) -th work steps included in each sub-process at the current time t ₀ {X _{i, j} (k): 1 ≦ i ≦ n, 1 ≦ j ≦ s (i)} (where j is a natural number), necessary for work in each work process included in each sub-process Work time {d _{i, j} : 1 ≦ i ≦ n, 1 ≦ j ≦ s (i) } And the time limit {r (k + m): 1 ≦ m ≦ N} (where m and N are natural numbers) when the work result in the k + 1,... In the computer program for creating a work schedule in the work process, by selecting the sub-process to be entered and determining the work time input time by the max-plus algebra,
When the i-th sub-process is selected as the sub-process for performing the k + m-th work on the computer, δ _i (k + m) = e (where e = 0), and the i-th sub-process is the sub-process for performing the k + m-th work. When a sub-process other than a process is selected, selection parameters {δ _i (k + m): 1 ≦ i ≦ n, 1 ≦ m ≦ N} satisfying δ _i (k + m) = ε (where ε = −∞) A selection parameter determination procedure for determining a value;
An N-row, one-column matrix T ₀ in which the value of each component is t ₀ , an N-row, one-column matrix R (k + 1) given by the following equation,
R (k + 1) = [r (k + 1) r (k + 1)... R (k + m)] ^T
(S (1) +... + S (n)) row and column matrix x (k) given by the following equation:
x (k) = [x _1,1 (k) x _1,2 (k) ... x _{1, s (1)} (k) x _2,1 (k) ...
x _{2, s (2)} (k) ... x _{n, 1} (k) ... x _{n, s (n)} (k)] ^T
It contains max-plus algebraic multiplication (circle x) and addition (circle +), and is given by the following equation containing a matrix A _i 'of s (i) rows s (i) columns (s (1 ) +... + S (n)) row (s (1) +... + S (n)) column matrix A _k ,

(S (1) +... + S (n)) row and column matrix B _k given by the following equation:

(S (i) -1) ε components and d _{i, s (i)} and δ _i (k) are given by the following formulas alternately including components multiplied by the multiplication of the max-plus algebra. One row (s (1) +... + S (n)) column matrix C _k ,

A matrix Γ _k given by

And a matrix Δ _k given by

a procedure for calculating the time point {u (k + m): 1 ≦ m ≦ N} when the k + 1,..., k + Nth work object is input by the following formula including the Θ operation of the max-plus algebra;

A computer program comprising: The selection parameter determination procedure includes:
In the computer, the value of the evaluation function including the penalty for delaying the scheduled delivery time of the work result determined by determining the value of the selected parameter and the work cost required for the work is minimized. To determine the value of the selected parameter
The computer compares the lower bound value of the evaluation function obtained from the selection parameter with a fixed value and the provisional minimum value, which is the minimum value of the evaluation function provisionally obtained, to determine the value of the selection parameter. The method includes a step of searching for a combination of selection parameter values that gives a minimum value of the evaluation function from among n ^N combinations of selection parameter values by a branch and bound method for selecting combinations. The computer program according to 8. A procedure for causing the computer to calculate a time point u (k + 1) at which the (k + 1) -th work object is input according to the following formula including a matrix [e ε.

A procedure for causing a computer to calculate a matrix x (k + 1) obtained by the following equation;

A procedure for causing the computer to calculate x (k + 1) as new x (k);
Causing the computer to update the current time t ₀ and the deadline {r (k + m): 1 ≦ m ≦ N};
The computer obtains the above-mentioned selection parameters obtained by setting δ _i (k + m + 1) as a new δ _i (k + m) (where 1 ≦ m ≦ N−1) and δ _i (k + N) as a new δ _i (k + N). A procedure for setting the value of the evaluation function as an initial value of the provisional minimum value;
The computer program according to claim 9, further comprising: causing the computer to create a schedule again.

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