CN111667113A

CN111667113A - Rapid deadlock-free branch-bound scheduling method for flexible manufacturing system

Info

Publication number: CN111667113A
Application number: CN202010497896.4A
Authority: CN
Inventors: 罗建超
Original assignee: Northwestern Polytechnical University
Current assignee: Northwestern Polytechnical University
Priority date: 2020-06-04
Filing date: 2020-06-04
Publication date: 2020-09-15
Anticipated expiration: 2040-06-04
Also published as: CN111667113B

Abstract

The invention provides a quick deadlock-free branching and delimiting scheduling method of a flexible manufacturing system, which is based on flexible manufacturing system modeling of a position timing Petri network, searches for optimal scheduling in a branch tree of a PNS, gives a node which is not branched, cuts off no promising node in the searching process, and continuously updates the completion time of scheduling until all nodes are branched or the CPU time is more than the set maximum CPU time. The method does not need to spend time for parameter adjustment, can avoid repeated searching, and has high searching efficiency; since the scheme repeatedly selects nodes with small lower bounds from the top layer to the bottom layer for branching, a satisfactory solution can be given in a short time, and the more time is given, the higher the quality of the found solution is.

Description

Rapid deadlock-free branch-bound scheduling method for flexible manufacturing system

Technical Field

The invention relates to the field of flexible manufacturing, in particular to a branch and bound scheduling method.

Background

Flexible manufacturing systems are widely used in industrial processes such as semiconductor manufacturing, mail systems, eyewear production, and the like. Due to the presence of shared resources in a flexible manufacturing system, system deadlocks may result when a workpiece requests a shared resource. Once the deadlock occurs, the system is in a halt and cannot complete the processing task, which causes economic loss and serious consequences. Scheduling is the basis for ensuring the performance of the system. The deadlock-free scheduling method not only ensures that the system is deadlock-free, but also ensures the operation performance of the system. Therefore, it is more complicated than the deadlock free control problem of the system. Recently, the problem of deadlock free scheduling of flexible manufacturing systems has received a great deal of attention from researchers. Xing et al, described in document [ 3 ] (k.y.xing, l.b.han, m.c.zhou, and f.wang, "delay-free genetic scheduling algorithm for automated manufacturing systems based Deadlock control policy," IEEE trans.syst., Man, cybern., B, cybern., vol.42, pp.603-615, jun.2012.) integrate Deadlock avoidance strategies into genetic algorithms, propose a Deadlock free genetic scheduling algorithm, and propose a detection algorithm of chromosomes and a repair algorithm for repairing non-viable chromosomes to viable chromosomes. Luo et al in document [1 ] (J.C.Luo, K.Y.Xing, M.C.ZHou, X.L.Li, and X.N.Wang, "delay-free scheduling of automated manufacturing systems using Petri nets and dhybrid statistical search," IEEE trans.Syst., Man, Cybern. To narrow the search space, they constrain the search process to a dynamic window. In document [ 4 ] (O.T. Baruwa, M.A. Piera, and A.Guasch, "Deadlock-free scheduling method for flexible manufacturing systems based on timed coordinated Petri nets and analytical search," IEEE trans. Syst., Man, Cybern., Syst., vol.45, No.5, pp.831-846, May.2015.), Baruwa et al model the system using colored Petri nets, a hierarchical search algorithm was proposed. The result shows the effectiveness of the algorithm through the test of the standard case. In document [ 2 ] (x.n.wang, k.y.xing, x.l.li, and j.clu, "analysis of distribution algorithm for scheduling protocol of flexible manufacturing systems using Petri nets," application.math.model., vol.55, pp.776-788, ma.2018.), Wang et al propose a system-based Petri deadlock net model, without distributed estimation scheduling algorithm. By embedding the local search algorithm into the distributed estimation algorithm, the search capability of the algorithm is improved.

The current prior art has the following disadvantages:

1) the parameter adjustment is difficult, the scheduling performance depends on the parameter adjustment, and the parameter setting is too much and irregular, so that the parameter adjustment is difficult;

2) the searching efficiency is low, random searching is taken as the main point, and good scheduling still cannot be obtained within a long searching time.

Disclosure of Invention

In order to overcome the defects of the prior art, the invention provides a quick deadlock-free branch-and-bound scheduling method of a flexible manufacturing system. Firstly, a lower bound of the completion time of the nodes is established based on the reachable graph of the system so as to evaluate the quality of the nodes. And secondly, selecting a branch node according to the level of the node and the lower bound of the node, and obtaining the level of the next selected node. Thirdly, according to the lower bound and the level of the nodes, a pruning rule is established, and repeated searching is avoided. Fourthly, the optimal deadlock avoidance strategy is embedded into the searching process, and the fast deadlock-free branch delimiting method is established. The invention enables fast obtaining of better scheduling than the most advanced methods available.

The technical scheme adopted for solving the technical problems comprises the following specific steps:

step 1: flexible manufacturing system modeling

Step 1.1Petri Net preparation knowledge

The Petri Net is a directed bipartite graph comprising vertices of two types, location (place) and transition (transition), where locations are represented by circles, transitions are represented by rectangles or thick solid lines, locations and transitions are connected by directed arcs,

a set of non-negative integers is represented,

a set of positive integers is represented that,

represents a set of positive integers {1, 2.., k }; the Petri net is specifically defined as follows:

a Petri net is a 3-tuple of N ═ (P, T, F), where P is the set of locations, T is the set of transitions,

for directed arc sets, P and T are finite disjoint non-empty sets, i.e.

In the Petri net N ═ (P, T, F), a vertex x ∈ P ∪ T is given, and^·x ═ y ∈ P ∪ T | (y, x) ∈ F } and x^·Y ∈ P ∪ T | (x, y) ∈ F represents a preamble set (preset) and a postset (postset) of x, respectively;

in a Petri net N ═ (P, T, F), a path (path) is a sequence α ═ x composed of adjacent vertices₁,x₂,...,x_kWherein x is_i∈P∪T，(x_i,x_i+1)∈F，i∈Z_k-1；

The state or identity (marking) of a Petri net N ═ P, T, F is a mapping M: P → Z from a set of positions to a set of non-negative integers^*For position P ∈ P and marker M, M (P) represents the number of tokens in position P under marker M, the number of tokens in position being represented by a black dot number or a positive integer, with initial marker M₀The Petri net N is called an identification Petri net and is marked as (N, M)₀)；

The identification is represented by a vector, i.e. M ═ M (p)₁),M(p₂),...,M(p_|P|))^TFor brevity, ∑_p∈PM (p) p represents the identifier M;

given a Petri net N ═ (P, T, F), T ∈ T is a transition in N, M is an identification in N, if

M(p)>0, then the transition t is called enabled (enabled) under the identifier M, and is recorded as M [ t >; triggering a (fire) transition t under the identity M causes the system to transition from the identity M to the identity M ', denoted M [ t > M', where,

M′(p)＝M(p)-1；

m' (p) ═ M (p) + 1; otherwise, M' (p) ═ M (p); given a controller Δ, let M [ t, Δ > denote that initiating an enable transition t at an identification M is permitted by the controller Δ, enabling for a transition sequence τ t₁t₂...t_k，t_i∈T，i∈Z_kIf M is_i[t_i＞M_i+1Wherein M is₁Let τ be feasible under M (feasible);

(N₁,M₁₀)＝(P₁,T₁,F₁,M₁₀) And (N)₂,M₂₀)＝(P₂,T₂,F₂,M₂₀) Is two labeled Petri nets, if

M₁₀(p)＝M₂₀(p) is called (N)₁,M₁₀) And (N)₂,M₂₀) Is compatible; two compatible labeled Petri nets (N)₁,M₁₀) And (N)₂,M₂₀) The composition (composition) of (A) is a new identification Petri net

Wherein P ═ P₁∪P₂；T＝T₁∪T₂；F＝F₁∪F₂；

M₀(p)＝M₁₀(p)，

M₀(p)＝M₂₀(p)；

For the logo Petri Net (N, M)₀)＝(P,T,F,M₀) Let B (N, M)₀)＝(V, E) represents (N, M)₀) Wherein V is (N, M)₀) E is a set of directed edges between nodes; a node consists of a transition sequence; let v₁And v₂Respectively indicates that two nodes satisfy M₀[v₁＞M₁And M₀[v₂＞M₂If T ∈ T is present, so that M₁[t>M₂Then there is a slave v₁To v₁Directed arc of, in a branch tree, node v₁Is of depth | v₁L +1, where v₁| represents v₁The number of medium transitions; all nodes at the same depth form a search layer, and the layer with the largest depth is called a bottom layer (bottom level);

step 1.2 Flexible manufacturing System modeling based on position-timed (planned-timed) Petri Net

Assuming that the resource type and the workpiece type in the flexible manufacturing system are m and n, respectively, the resource set and the workpiece set are represented as m and n, respectively

And

given resource r_i∈ R, order

Indicates its initial resource amount, order

Indicating the resource instance set, initially the number of unprocessed u-th class artifacts is Ψ_u；

An unprocessed workpiece is processed sequentially on a series of resources, the processing sequence of which is called a path, such that

Means all ofA set of machining paths, wherein the u-th class workpiece has a plurality of machining paths; order to

Set of machining paths representing class u workpieces, let L_iDenotes theta_iNumber of medium resources, then theta_iIs shown as

Wherein

Is theta_iThe j-th resource in (c); let a_ijIndicates a certain value at theta_iJ-th movement of the workpiece to be machined, theta_iCorresponding to a sequence of activities, i.e.

Resource r_k∈ R is requested by multiple activities

Indicates all requests r_kThe activity of (2); for theta_uTwo paths θ in (b)_iAnd theta_jIf, if

Then activity a_ikAnd a_jkAre the same; for the sake of simplicity, a_ikAnd a_jkAre combined into one activity, the set of all activities is represented as

Given a_ij∈A(r_k) Let path t_ijp_ijt_i(j+1)Representing its Petri Net model, where t_ijDenotes a_ijStart of) operation, t_i(j+1)Denotes a_ijIs performed by p_ijDenotes a_ijThe active position of (a); resource r_k∈ R resource location R_kIs represented by_kThe initial token number of C_kAdding two arcs, one from r_kTo t_ijRepresenting a request r_kThe other one from t_i(j+1)To r_kRepresents the release of r_kLet OPN_ijDenotes a_ijThe Petri Net model of (1);

for class u workpieces, two imaginary active positions p are added_u◇And p_u◆Respectively representing the start and end of the motion of the u-th workpiece, for theta_i∈Θ_uAdding two arcs, one from p_u◇To t_i1And the other is from

To p_u◆The overall flexible manufacturing system is modeled as:

wherein P is a set of active positions;

is a resource location set; t is a transition set; f is a set of arcs, M₀Is an initial mark and is expressed as

M₀(p_u◇)＝Ψ_u；

M₀(p)＝0；

M₀(r_i)＝C_i；

For P ∈ P, let

Representing the processing time of the activity associated therewith, for

d (p) ═ 0, the Petri network with position assignment is called scheduling Petri network (PNS for short); let M_fRepresents the termination identifier of N, wherein

M_f(p_q◆)＝Ψ_q；

M_f(p)＝0；

M_f(r_i)＝C_iOne feasible schedule is a transition sequence of π and satisfies M₀[π＞M_fLet S (N, M)₀) Represents (N, M)₀) A set of feasible schedules;

step 2: deadlock-free fast branch-and-bound scheduling algorithm

Based on a Petri network model of a system, a deadlock-free rapid branching and delimiting scheduling algorithm is provided, an optimal solution is searched in a branch tree of the system, the searching process is guided by a new branching strategy, the branching strategy overcomes the defects of breadth-first searching and depth-first searching, the lower bound of a node is provided for pruning nodes which are not promising, and in addition, two pruning rules are provided for avoiding repeated searching;

step 2.1, lower bound of the node;

let v equal t₁t₂...t_qRepresents B (N, M)₀) And M is one node in₀[ v > M, [ tau (v) denotes t_qInitiation time of l^*(v)＝min{τ(vv₁)|v₁∈{v₂|M[v₂＞M_f}; if present

And l is less than or equal to l^*(v) Then l is the lower bound of v; if the lower bound of node v is equal to l^*(v) Then the lower bound is the maximum lower bound for v; l^*(v) A minimum value equal to the maximum value of the completion time of the resource from M to complete the processing of all the remaining workpieces;

order to

Representing the first token or workpiece in p, assuming the workpiece

The time for entering p is tau (v)₁) Then, at time τ (v),

the remaining time in p is

A path is said to be a machining path if all positions in the path are active, given that P '∈ P (P' ≠ P) satisfies the existence of a machining path ω ═ pt₁p₁t₂…p_k-1t_kp_k(p_kP'), order

{ D (ω) | ω is the processing path from p to p';

given T ∈ T, let^(a)table with t ═ pAn input active position showing t, ζ (t, v) representing the fastest time to complete the workpiece from τ (v) in position p; if M (p) is not less than 1

If M (p) is 0, let

Machining path

And is

If it is not

Then

If not, let ξ (r ∞)_iV) denotes r starting from τ (v)_iThe fastest time to request, ξ (r)_i,v)＝min{ζ(t,v)|t∈r_i ^·}；

Given r_ik∈Ξ_iLet I (r)_ikV) denotes that r is scheduled from τ (v) to the next activity_ikR between_ikA minimum idle time of; if r is_ikIs idle at τ (v), then I (r)_ik,v)＝ξ(r_iV) - τ (v); otherwise

Wherein

Is being represented by r at τ (v) in q_ikJ-th activity of processing; let gamma (r)_ikV) indicates that the next activity is scheduled at r_ikAt the earliest time of (i), then gamma (r)_ik,v)＝τ(v)+I(r_ik,v)+χ(r_ikV), wherein if r_ikAt τ (v) is idle, then χ (r)_ikV) is 0; if at time r (v)_ikIs being processed

Then

The path selection scheme assigns a processing path to each unfinished workpiece; let S (v) ═ s₁,s₂,...,s_hDenotes from M to M_fGiven s ∈ S (v), let A (s, r)_i) Indicating routing to r by the routing scheme_iGiven activity a_jk∈A(s,r_i) Let p (a)_jk) Indicates its active position, order

Xi when representing path selection scheme as s_iMinimum average processing time for medium resource instances;

definition 1: given B (N, M)₀) Node v in (1), define

Theorem 1: given B (N, M)₀) Is connected to the node v in (1),

lower bound of completion time of v, i.e.

Step 2.2 defining branching strategies

Let d denote the depth of the current search level, d denotes the maximum depth of the search tree, at the search level d, the branching policy preferentially selects the node with the smallest lower bound, if there is no node that is not branched at the search level d or d ═ d, then set d ═ d + 1)% d, and reselect the node; stopping selecting a node if all depths are searched and no nodes not branched are selected; otherwise, one node not branched is selected, d ═ d + 1)% d is set, and the node is reselected;

the branching of the selected node v is directed by the deadlock controller, assuming M₀[ v > M, only transitions enabled under M and allowed by the controller can initiate and produce new unbranched nodes; vector quantity

And

are used to store branched and unbranched nodes respectively,

and

for storing lists of nodes at depth i that are branched and unbranched, respectively; the steps of the branching strategy are as follows:

inputting: depth d of the current search layer;

and (3) outputting: flag, d, v and list, wherein, flag true represents that a node which is not branched is selected, and flag false represents that a node which is not branched is selected; v is the selected non-branched node; list is used for storing newly generated nodes which are not branched;

initialization: d₁＝d，

flag＝true，v＝v₀Wherein v is₀Is an empty transition sequence;

step 2.2.1: when d is₁D ═ or

When the collection is empty, the circulation of the steps 2.2.1.1 and 2.2.1.2 is carried out, and the following steps are specifically carried out:

step 2.2.1.1: let d₁Is (d)₁+1)％d*；

Step 2.2.1.2: if d is₁If d is equal, the flag is made false, and the loop is ended, otherwise, the step 2.2.1 is skipped;

step 2.2.2: if flag is true, then:

step 2.2.2.1: from

In selection

Minimum node v₁；

Step 2.2.2.2: v is to be₁From

Removing;

step 2.2.2.3: let M₀[v₁> M, let e (M) { T ∈ T | M [ T, PCF >);

step 2.2.2.4 for any change t ∈ E (M) { v₁t } adding to list;

step 2.2.2.5: let v equal v₁Where v is the selected unbranched node;

step 2.2.2.6: let d be (d)₁+1)％d*；

Step 2.2.3: returning flag, d, v and list;

step 2.3 definition of pruning rules

Once one isAn unbranched node is selected and branched to be inserted into

In the method, a new node generated by a branch is inserted into

In the method, two pruning rules are proposed, which are respectively applied to inserting nodes into the tree

And

in the process of (2):

pruning rule 1

Inputting: a branched node v;

initialization: let M₀[v＞M；d＝|v|+1；

Step A-1: if present

So that M₀[v₁If the value is more than M, executing the step A-1.1, otherwise executing the step A-2;

step A-1.1: if it is not

Then order

Is composed of

Finishing;

step A-2: order to

Is composed of

Pruning rule 2

Inputting: a newly generated unbranched node v;

initialization: let M₀[v＞M；d＝|v|+1；

Step B-1: if present

So that M₀[v₁If the value is more than M, executing a step B-1.1, otherwise executing a step B-2;

step B-1.1: if it is not

Then order

Is composed of

Finishing;

step B-2: if present

So that M₀[v₁If the value is more than M, executing a step B-2.1, otherwise executing a step B-3;

step B-2.1: if it is not

Then order

Is composed of

Order to

Is composed of

Finishing;

step B-3: order to

Is composed of

Step 2.4, a deadlock-free rapid branching and delimiting algorithm;

deadlock free fast Branch bounding method (Anytime Branch and Bound, hereinafter referred to as AB&B) Is a search algorithm for searching the optimal schedule in the branch tree of PNS

Indicating the current best solution completion time. Given an unbranched node v, if

Then v is an unprivileged (unprivising) node because branch v cannot get a better solution than the current best solution. During the searching process, nodes which are not promising can be cut off;

at the beginning of the process, the process is carried out,

using a branching strategy, a node is selected and branching is performed under the supervision of PCF, and the selected node is inserted under the supervision of a pruning strategy 1

Under supervision of pruning strategy 2, newly generated nodes are inserted into

If a completion time is found to be less than

Will then

Updated as the completion time of the schedule, once

Is updated at

Middle lower bound is greater than or equal to

The nodes are pruned, and the process is carried out until all the nodes are branched or the CPU time is more than the set maximum CPU time; the detailed steps of the deadlock-free fast branch-and-bound algorithm are as follows:

inputting: maximum CPU time

And (3) outputting: current best schedule v;

initialization:

wherein v is₀Is an empty transition sequence;

order to

Let v be v₀(ii) a Let the current search layer depth d be 1;

step 2.4.1: if the CPU

Circularly executing the steps 2.4.1.1-2.4.1.9, otherwise, jumping to the step 2.4.2;

step 2.4.1.1: let the input of the branching strategy be d, and execute the branching strategy, let the result of the branching strategy output be (flag, d, v, list);

step 2.4.1.2: if flag is false, jumping to step 2.4.2;

step 2.4.1.3: let the input of pruning rule 1 be v, execute pruning rule 1;

step 2.4.1.4: for all v₁∈ list, circularly executing the step 2.4.1.5-2.4.1.8;

step 2.4.1.5: let the input of pruning rule 2 be v₁Executing pruning rule 2;

step 2.4.1.6: if | v₁And | +1 ═ d-

Then steps 2.4.1.7 and 2.4.1.8 are performed, otherwise go to step 2.4.1.4;

step 2.4.1.7: order to

Is tau (v)₁) (ii) a Let v be v₁；

Step 2.4.1.8: from

Deletion in

And satisfy

All v of₂；

Step 2.4.1.9: skipping to step 2.4.1;

step 2.4.2: v is returned.

The invention has the beneficial effects that:

(1) because the deadlock-free branch delimitation method only comprises one parameter (the maximum CPU time) and the maximum CPU time can be set according to the actual requirement, the scheme does not need to spend time for parameter adjustment;

(2) the method only keeps the node with the minimum lower bound in the nodes with the same identification reached by the system, so the scheme can avoid repeated searching. In addition, the scheme replaces random search with regular search, so that the method has high search efficiency;

(3) since the scheme repeatedly selects nodes with small lower bounds from the top layer to the bottom layer for branching, a satisfactory solution can be given in a short time, and the more time is given, the higher the quality of the found solution is.

Drawings

FIG. 1 is a PNS model of the flexible manufacturing system of the present invention.

FIG. 2 is a graph of node mean lower bound and

the difference of (a).

Detailed Description

The invention is further illustrated with reference to the following figures and examples.

The innovation points of the invention are as follows:

(1) the deadlock-free branch delimitation method only comprising one parameter (maximum CPU time) is provided, when the allowable time is shorter, a satisfactory solution can be obtained, and when more time is given, a better solution can be obtained;

(2) the lower bound of the completion time of the search node is provided, the nodes with small lower bound of the node are preferentially selected to branch in the same search layer, and the random search is replaced by the regular search;

(3) a new branching strategy is provided, and nodes with small lower bound are repeatedly selected from the top layer to the bottom layer for branching, so that a large amount of time is avoided from spending in some poor search areas;

(4) and (3) a pruning rule is provided, and only the node with the minimum lower bound in the nodes with the same identification is kept, so that repeated searching is avoided, and the searching efficiency of the algorithm is improved.

The method comprises the following steps:

step 1: flexible manufacturing system modeling

Step 1.1Petri Net preparation

a set of non-negative integers is represented,

a set of positive integers is represented that,

for directed arc sets, P and T are finite disjoint non-empty sets, i.e.

The identification is represented by a vector, i.e. M ═ M (p)₁),M(p₂),...,M(p_|P|))^TFor brevity, ∑_p∈PM (p) p represents the identifier M; for example, P ═ { P ═ P₁,p₂,...,p₆When M is (0,1,0,0,0,2), the symbol M is denoted as p₂+2p₆；

M(p)>0, then the transition t is said to be enabled under the M identifier, denoted as M [ t>(ii) a Triggering a (fire) transition t under the identity M will cause the system to transition from the identity M to the identity M', denoted as M [ t ]>M', wherein,

M′(p)＝M(p)-1；

Wherein P ═ P₁∪P₂；T＝T₁∪T₂；F＝F₁∪F₂；

M₀(p)＝M₁₀(p)，

M₀(p)＝M₂₀(p)；

For the logo Petri Net (N, M)₀)＝(P,T,F,M₀) Let B (N, M)₀) (V, E) represents (N, M)₀) Wherein V is (N, M)₀) E is a set of directed edges between nodes; a node consists of a transition sequence; let v₁And v₂Respectively indicates that two nodes satisfy M₀[v₁＞M₁And M₀[v₂＞M₂If T ∈ T is present, so that M₁[t＞M₂Then there is a slave v₁To v₁Directed arc of, in a branch tree, node v₁Is of depth | v₁L +1, where v₁| represents v₁The number of medium transitions; all nodes at the same depth form a search layer, and the layer with the largest depth is called a bottom layer (bottomlevel);

The Petri Net model of the flexible manufacturing system consists of the processing path of the workpiece, and can also consist of Petri Net models of all activities/operations. Assuming that the resource type and the workpiece type in the flexible manufacturing system are m and n, respectively, the resource set and the workpiece set are represented as m and n, respectively

And

given resource r_i∈ R, order

Indicates its initial resource amount, order

Representing a set of all machining paths, wherein the u-th workpiece has a plurality of machining paths; order to

Wherein

Resource r_k∈ R is requested by multiple activities

To p_u◆The overall flexible manufacturing system is modeled as:

wherein P is a set of active positions;

is a resource location set; t is transitionGathering; f is a set of arcs, M₀Is an initial mark and is expressed as

M₀(p_u◇)＝Ψ_u；

M₀(p)＝0；

M₀(r_i)＝C_i；

For P ∈ P, let

Representing the processing time of the activity associated therewith, for

d (p) ═ 0, and such a place-assigned Petri net is called a scheduling Petri net (PNS for short); let M_fRepresents the termination identifier of N, wherein

M_f(p_q◆)＝Ψ_q；

M_f(p)＝0；

M_f(r_i)＝C_iOne feasible schedule is a transition sequence of π and satisfies M₀[π＞M_fLet S (N, M)₀) Represents (N, M)₀) The object of the invention is to set up a feasible scheduling in S (N, M)₀) Searching for a feasible schedule with the minimum completion time;

for example, consider a flexible manufacturing system developed in the literature [1, 2,3 ], the resource set of which is R ═ m₁,m₂,m₃,m₄,r₁,r₂,r₃}, the system processes three types of workpieces, i.e.

Θ＝{θ₁,θ₂,θ₃,θ₄In which θ₁＝r₂m₂r₂,θ₂＝r₁m₁r₂m₂r₃，θ₃＝r₁m₃r₂m₄r₃，θ₄＝r₃m₄r₂m₃r₁。Θ₁＝{θ₁}，Θ₂＝{θ₂,θ₃}，Θ₃＝{θ₄}。A(r₁)＝{a₂₁,a₄₅}，A(r₂)＝{a₁₁,a₁₃,a₂₃,a₃₃,a₄₃}，A(r₃)＝{a₂₅,a₄₁}，A(m₁)＝{a₂₂}，A(m₂)＝{a₁₂,a₂₄}，A(m₃)＝{a₃₂,a₄₄}，A(m₄)＝{a₃₄,a₄₂}. The Petri Net model of the system is shown in FIG. 1, with initial identification and processing time of the activity not given.

Step 2: deadlock-free fast branch-and-bound scheduling algorithm

A deadlock-free rapid branching and delimiting scheduling algorithm is provided based on a Petri network model of a system, and the algorithm searches an optimal solution in a branching tree of the system. The search process is guided by a new branching strategy that overcomes the shortcomings of breadth-first and depth-first searches. In order to prune off the nodes that are not promising, a lower bound of nodes is proposed. In addition, two pruning rules are proposed to avoid duplicate searches.

Step 2.1, lower bound of the node;

let v equal t₁t₂...t_qRepresents B (N, M)₀) One of (1)Node and M₀[ v > M, [ tau (v) denotes t_qInitiation time of l^*(v)＝min{τ(vv₁)|v₁∈{v₂|M[v₂＞M_f}; if present

And l is less than or equal to l^*(v) Then l is the lower bound of v. The larger the lower bound, the more pronounced the pruning effect. If the lower bound of node v is equal to l^*(v) Then the lower bound is the maximum lower bound for v; l^*(v) A minimum value equal to the maximum value of the completion time of the resource from M to complete the processing of all the remaining workpieces; to obtain a high quality (large) lower bound, the completion time of the resource must be accurately calculated. Since the resources are either in a processing state or an idle state, processing time and idle time of the resources need to be considered.

Order to

Representing the first token or workpiece in p, assuming the workpiece

The time for entering p is tau (v)₁) Then, at time τ (v),

the remaining time in p is

A path is said to be a machining path if all positions in the path are active, given that P '∈ P (P' ≠ P) satisfies the existence of a machining path ω ═ pt₁p₁t₂…p_k-1t_kp_k(p_k＝p')，Order to

{ D (ω) | ω is the processing path from p to p';

given T ∈ T, let^(a)t ═ p denotes the input active position of t, ζ (t, v) denotes the fastest time to complete the workpiece from τ (v) in position p; if M (p) is not less than 1

If M (p) is 0, let

Machining path

And is

If it is not

Then

Wherein

Then

definition 1: given B (N, M)₀) Node v in (1), define

Theorem 1: given B (N, M)₀) Is connected to the node v in (1),

lower bound of completion time of v, i.e.

And (3) proving that: let r be_ij∈Ξ_i，

d₁(v,v^*,r_ij)(d₂(v,v^*,r_ij) From τ (v) to τ (vv)^*) R between_ijIdle time (machining time) of, then d₁(v,v^*,r_ij)+d₂(v,v^*,r_ij)＝τ(vv^*)-τ(v)，

Hypothesis and v^*The corresponding path selection scheme is s^*Then, then

Due to the fact that

And is

Therefore, it is

Therefore, the temperature of the molten metal is controlled,

step 2.2 defining branching strategies

The branching strategy decides which node to select for branching and how to branch, and conventional methods such as breadth-first and depth-first can be applied to decide which node to select for branching, the number of nodes in the branching tree growing exponentially with the size of the scale of the problem. For a large scale problem, it is not possible to enumerate all nodes to solve the problem. Thus, the breadth first strategy is not suitable for large scale problems. The depth first strategy always finds the solution to the problem quickly. However, the depth-first strategy has two drawbacks. First, since it preferentially selects node branches with greater depth, it is likely to be at B (N, M)₀) Takes a lot of time. Second, if B (N, M)₀) The path of the medium optimal solution is longer, it may choose a much worse and long path before finding the optimal solution.

The invention provides a new branching strategy, which can quickly find the solution of the problem and overcome the defect of the depth priority strategy. Let d denote the depth of the current search level, d denotes the maximum depth of the search tree, at the search level d, the branching policy preferentially selects the node with the smallest lower bound, if there is no node that is not branched at the search level d or d ═ d, then set d ═ d + 1)% d, and reselect the node; stopping selecting a node if all depths are searched and no nodes not branched are selected; otherwise, one node not branched is selected, d ═ d + 1)% d is set, and the node is reselected; this strategy differs from the depth-first strategy in that it re-selects nodes from level 1 if d x level is reached, whereas the depth-first strategy restarts from level (d x-1). Such a difference makes it possible to overcome the two disadvantages of the depth-first strategy. A comparison of this strategy with the depth-first search strategy can be found in the next section.

The branching of the selected node v is directed by the deadlock controller, assuming M₀[vOnly transitions enabled under M and allowed by the controller can initiate and produce new unbranched nodes. Due to PNS and S³PR structures are essentially identical, so in document [ 5 ] (J.Ezpeleta, J.M.Colom, and J.Martinez, "A Petri net based on derived delivery polarity for Flexible manufacturing systems," IEEETrans.Robot.Autom., vol.11, No.2, pp.173-184, Apr.1995.) and document [ 6 ] (L.Piroddi, R.Cordone, and I.Fumagalli, "Selective Synthesis for derived delivery in Petri," IEEE.Sy., Man, cybern., A, Syst, Humans, vol.38, 1347-pp.8, Nov.2008), for S³The deadlock control method of PR may be used for PNS. Since the controller in the document [ 6 ], denoted PCF (consisting of the initials of the last name of the author), is the most permissive in existence, PCF is embedded in the branching strategy to avoid deadlock. Vector quantity

And

are used to store the branched and unbranched nodes, respectively.

And

for storing a list of nodes at depth i that are branched and not branched, respectively. The steps of the complete branching strategy are as follows:

inputting: depth d of the current search layer;

and (3) outputting: flag, d, v, and list, where flag true (false) indicates that there are (or no) nodes that are not branched selected; v is the selected non-branched node; list is used for storing newly generated nodes which are not branched;

initialization: d₁＝d，

flag＝true，v＝v₀Wherein v is₀Is an empty transition sequence;

step 2.2.1: when d is₁D ═ or

step 2.2.1.1: let d₁Is (d)₁+1)％d*；

step 2.2.2: if flag is true, then:

step 2.2.2.1: from

In selection

Minimum node v₁；

Step 2.2.2.2: v is to be₁From

Removing;

step 2.2.2.3: let M₀[v₁> M, let e (M) { T ∈ T | M [ T, PCF >);

step 2.2.2.4 for any change t ∈ E (M) { v₁t } adding to list;

step 2.2.2.5: let v equal v₁Where v is the selected unbranched node;

step 2.2.2.6: let d be (d)₁+1)％d*；

Step 2.2.3: returning flag, d, v and list;

step 2.3: defining pruning rules

Once an unbranched node is selected and branched, it is inserted into

In the method, a new node generated by a branch is inserted into

In (1). Because of the slave M₀The sequence of transitions that trigger may arrive at the same identity, so two different nodes may correspond to the same identity. The search process from nodes corresponding to the same identity is identical. To avoid repeated searches, two pruning rules are proposed, one for each node inserted into the tree

And

in the process of (1).

Pruning rule 1:

inputting: a branched node v;

initialization: let M₀[v＞M；d＝|v|+1；

Step A-1: if v is present₁∈B_dSo that M₀[v₁If the value is more than M, executing the step A-1.1, otherwise executing the step A-2;

step A-1.1: if it is not

Then order B_dIs (B)_d\{v₁}) ∪ { v }, end;

step A-2: let B_dIs B_d∪{v}；

Pruning rule 2

Inputting: a newly generated unbranched node v;

initialization: let M₀[v＞M；d＝|v|+1；

Step B-1: if v is present₁∈B _dSo that M₀[v₁If the value is more than M, executing a step B-1.1, otherwise executing a step B-2;

step B-1.1: if it is not

Then orderB _dIs prepared from (a)B _d\{v₁}) ∪ { v }, end;

step B-2: if v is present₁∈B_dSo that M₀[v₁If the value is more than M, executing a step B-2.1, otherwise executing a step B-3;

step B-2.1: if it is not

Then orderB _dIs B _d ∪ { v }, let B_dIs B_d\{v₁And fourthly, ending;

step B-3: order toB _dIs composed ofB _d∪{v}；

Step 2.4, a deadlock-free rapid branching and delimiting algorithm;

fast Branch and Bound algorithm (Anytime Branch and Bound, hereinafter referred to as AB)&B) Is a search algorithm for searching for the optimal schedule in the branch tree of the PNS. Order to

Represents the current best solution completion time, given an unbranched node v, if

Then v is an unprivileged (unprivising) node because branch v cannot get a better solution than the current best solution. During the search, no promising nodes may be pruned.

At the beginning of the process, the process is carried out,

using a branching policy, a node is selected and branches under the supervision of the PCF. Under supervision of pruning rule 1, the selected node is inserted into

In (1). Under supervision of pruning rule 2, newly generated nodes are inserted into

Performing the following steps; if a time-out less than

Will then

Updating the time as the completion time of the scheduling; once the cover is closed

Is updated at

Middle lower bound is greater than or equal to

The process continues until all nodes are branched or the CPU time is greater than the set maximum CPU time. The detailed steps of the deadlock-free fast branch-and-bound algorithm are as follows:

inputting: maximum CPU time

And (3) outputting: current best schedule v;

initialization:B ₁＝{v₀in which v is₀Is an empty transition sequence;

order to

Let v be v₀(ii) a Let the current search layer depth d be 1;

step 2.4.1: if the CPU

step 2.4.1.2: if flag is false, jumping to step 2.4.2;

step 2.4.1.3: let the input of pruning rule 1 be v, execute pruning rule 1;

step 2.4.1.5: let the input of pruning rule 2 be v₁Executing pruning rule 2;

step 2.4.1.6: if | v₁And | +1 ═ d-

Then steps 2.4.1.7 and 2.4.1.8 are performed, otherwise go to step 2.4.1.4;

step 2.4.1.7: order to

Is tau (v)₁) (ii) a Let v be v₁；

Step 2.4.1.8: fromBDeletion of v₂∈BAnd satisfy

All v of₂；

Step 2.4.1.9: skipping to step 2.4.1;

step 2.4.2: v is returned.

Theorem 2: AB & B can generate a first schedule in polynomial time;

and (3) proving that: the PCF prohibits the initiation of transitions that cause deadlocks and all nodes are generated under the supervision of the PCF. This ensures that all slaves M₀The identity of the arrival of the transition sequence in the originating node is secure. That is, for any one generated tag unless it is M_fThere is at least one transition that can be initiated.

At the beginning of the process, the process is carried out,

when d is 1, the branching strategy selects v₀Under the supervision of PCF, the branch generates all son nodes thereof to be stored in list, and v is set as v ═ v₀And d is 2. Since there is at least one transition from M under the supervision of PCF₀Is initiated so that

Because of the fact that

And

is empty, so v is inserted under the supervision of pruning rule 1

And all nodes in list are inserted under the supervision of pruning rule 2

In (1). At d ═ 2, the branching strategy is selected from

In (b) selection v₁Under the supervision of PCF, the branch generates all son nodes thereof to be stored in list, and v is set as v ═ v₁And d is 3. Let M₀[v₁>M₁. Since there is at least one transition under the supervision of the PCF, the slave M can be used₁Is initiated so that

Because of the fact that

And

is empty, so v is under the supervision of pruning rule 1₁Is inserted into

And all nodes in list are inserted under the supervision of pruning rule 2

In (1). This process can be repeated until d-1. In d-1, the branching strategy is selected from

In (b) selection v₁Under the supervision of PCF, the branch generates all son nodes thereof to be stored in list, and v is set as v ═ v₁And d is 0. Let M₀[v₁>M₂. Since there is at least one transition under the supervision of the PCF, the slave M can be used₂Is initiated so that

Because of the fact that

And

is empty, so v is inserted under the supervision of pruning rule 1

And all nodes in list are inserted under the supervision of pruning rule 2

In (1). Given v₂∈ list, due to | v₂|+1＝d*，v₂Is the first found schedule. Thus AB&B can obtain the first schedule at the (d x-1) th iteration. Each node has at most | T | child nodes. Thus, the complexity of the branching strategy is O (| T |). At each of the iterations it is desired to,

and

are all empty. Thus, the complexity of applying pruning rules 1 and 2 is both O (1). Thus, AB&B has a complexity of O (| T |, d |);

AB & B can quickly find the first solution. And it can quickly update the current best solution, giving a reasonably good solution in a short time. It can give a better solution if it is given more time.

The experimental results are as follows:

AB & B is realized by C + +, is compiled by MSbuild 4.0, and runs on a personal computer with 1.7-GHz CPU and 4GB memory. The operating system of the computer is Windows 7 professional edition. AB & B was tested by a widely used flexible manufacturing system whose Petri net model is shown in fig. 1. The processing time of the campaign is specified in table 1.

TABLE 1 processing time of the campaign

Test results of lower bound, branching strategy and pruning rules

The first experiment was aimed at testing the validity of the lower bound, branching strategy and pruning rules. Suppose the number of three types of workpieces is 8, 12, 8, m₁-m₄And r₁-r₃Are 2, 2, 2, 2, 1, for this example, d 8 × 4+12 × 6+8 × 6+1 153 for lower bound validity detection at CPU times of 2s, 5s, 10s, 20s, 50s, 100s, 200s, 500s, and 1000s,

the average lower bound of nodes with a medium depth of 10, 20, 30.., 150 is recorded. At the same time, at these moments

And is also recorded. The results are shown in Table 2, the average lower bound of the nodes and

the difference of (c) is shown in fig. 2. As can be seen from Table 2, as the depth increases, the average lower bound of the node becomes closer to

This indicates that the greater the depth, the more accurate the estimation of the lower bound of the node. It is worth mentioning that the number of nodes at

depths

10 and 140 is 0 at CPU times of 100s, 200s, 500s, and 1000 s. This indicates that more and more nodes are at a relatively small depth ([1,10 ]]) And relatively large depth ([ d x-10, d x)]) Is pruned. As can be seen from FIG. 2, as the search progresses, the evaluation of the node is bounded by

The gap of (a) is getting smaller. This indicates that the proposed lower bound has a significant pruning effect on nodes that are not promising.

TABLE 2 average lower bound of nodes and

to test the effectiveness of the proposed branching strategy, the patent implements a depth-first strategy that is always on

And selecting the node with the maximum depth and the minimum lower bound for expansion. AB that will use this branching strategy&The B algorithm is represented as AB&B₁. To test the effectiveness of the proposed pruning rules, this patent implements AB without using pruning rules 1 and 2&B Algorithm, denoted AB&B₂。AB&B₂Directly inserting all branched and non-branched nodes into the nodes respectively

And

in (1). Maximum run time of all algorithms

Is set to 1200 s. The results of the runs are shown in Table 3.AB&B will improve the completion time of the current best schedule (C)_max) Spending 143.7s from 443 to 243. AB&B₁It takes 190.5s to improve the currently best scheduled completion time from 443 to 363. AB&B₂It takes 454.5s from 443 to 312 to improve the currently best scheduled completion time. Obviously, AB is the three algorithms&B is most efficient in searching. This indicates AB&The search efficiency of B has a great relationship with both branching strategies and pruning rules.

TABLE 3 scheduling results for AB & B, AB & B1 and AB & B2

Comparison with existing work

AB&B will be compared to all existing 6 methods, which are denoted M for the 6 algorithms respectively₁(method of reference [ 2 ]), M₂(method of reference [1 ]), M₃(method of reference [ 4 ]), M₄(method of reference [ 7 ]), M₅(method of reference [ 8 ]) and M₆(the method of document [ 9 ]), wherein document [ 7 ] is H.Lei, K.Y.Xing, Z.X.Gao, and F.L.Xiong, "A hybrid differential evaluation system" T.I.Meas.control, vol.38, No.10, pp.1270-1280,2016, document [ 8 ] is L.B.Han, K.Y.Xingn, X.Chen, and F.L.Xiong, "A hybrid differential evaluation system" 10, pp.1270-1280,2016, document [ 7 ] is L.B.Han, K.Y.Xingn, and F.L.Xiong, "A hybrid-based simulation evaluation system for scheduling analysis and analysis," L.F.F.Xingn. 1096,2018, and "L.Xingn.F.Gts and deadlocking controllers, "int.j.prod.res., vol.52, No.5, pp.1557-1572, oct.2014. the maximum run time of the setting algorithm is 1200 s. the testing is performed with 16 cases based on the criteria of fig. 1 in document [ 9 ]. in order to save space, the basic information of these standard cases is omitted, these cases are denoted as F1-16. for F16, the initial number of three types of workpieces is 20. therefore, for this example, d is 20 × 4+20 × 6+20 × 6+1 321. each node in the branching tree has at most 16 child nodes³²⁰(a rather bulky scale).

The results are shown in table 4, wherein other results are from documents [1, 2,4,7,8,9 ], respectively. In Table 4, BST represents the completion time of the current best solution, and AVG represents the average completion time of the current best solution found by multiple runs. AB&B all get the best solution better than the algorithm M₂，M₃And M₆。AB&B finds the 12 current best solutions (in bold) of the 16 examples. For 3 of the remaining 4 examples, AB&The best solution found by B is better than M₄。

Importantly, AB&B is stable in performance, while M₁，M₂And M₄-M₆The performance of (c) is random. For 15 of 16 examples, AB&The completion time of the best solution found by B is much less than the average completion time of the best solutions obtained by running the algorithms several times. Furthermore, M₁，M₂And M₄-M₆Depends on the fine-tuning of at least 3 parameters, and AB&B contains only one parameter, the maximum CPU time, which usually does not need to be adjusted. When the maximum CPU time is tight, AB&B is able to output a feasible and satisfactory solution. If more time is allowed, AB&B can be better and better solved.

The initial number of three types of workpieces in cases F3, F7, F11, and F15 was 15, 20, and 25, respectively. The initial number of three types of workpieces in cases F4, F8, F12, and F16 was 20, 20, and 20, respectively. For all these cases, AB & B found the best solution better than all the methods available. This indicates that AB & B has the ability to find a good solution to large scale problems. The superior capabilities of AB & B are attributed to lower bounds, branching strategies and pruning rules.

TABLE 4 calculation results of AB & B and the existing 6 methods

TABLE 5 AB&B，M₁，M₃And M₅Run the computer

TABLE 6 execution time of AB & B and existing algorithms

To find out if there is a difference between the best solutions found by these 7 algorithms, a parametric-free Friedman (Friedman) test was performed with a significance level of 0.05. The result of 90 indicates that AB&There are significant differences between B and these 6 algorithms. To be more specific, multiple comparisons were made at a significant level of 0.05 (described in [ 10 ] (M.Hollander and D.A.Wolfe, Nonparametric Statistical methods.New York: Wiley, 1973)), and the results indicated that AB was present at a significant level&B is significantly better than all existing algorithms. Furthermore, M₁And M₅Is obviously superior to M₂-M₄And M₆。

M₄And M₆Is not given in the document [ 9 ] and the document [ 7 ]. In the document [1 ], only M is given₂1900s and 178 s. Thus, only M₁，M₃And M₅An execution time comparison is performed. The execution time of the algorithm is affected by the speed of the CPU and the size of the memory. Generally, the faster the CPU runs and the larger the memory, the shorter the time to run the algorithm. AB&B，M₁，M₃And M₅The CPU and memory of the running computer of (a) are shown in table 5. It is clear that it is possible to use,AB&b runs the computer the slowest.

AB&B，M₁，M₃And M₅Execution time of (2) is shown in Table 6, where M₁，M₃And M₅The results are from documents [ 2,4,8 ], respectively. Note that only AB&B and M₃The time taken to find the first schedule is given. From Table 6, AB&B to M₃Finding the first schedule takes much less time. In addition, of the 16 examples, there are 12 examples AB&B takes the least time, there are 4 instances M₃The time spent is minimal.

To find AB&B，M₁，M₃And M₅Was tested at a significant level of 0.05 for a non-parametric friedman test. The results show that AB&B and M₁，M₃And M₅There is a significant difference between the execution times of (c). To be more specific, we made multiple comparisons at a significance level of 0.05. The results show AB&B spends the least time in these 4 algorithms. Due to AB&B is the slowest of the 4 algorithms to run, so AB&B has the highest computational efficiency.

To summarize:

the invention provides a fast branch and bound algorithm (AB & B) for the first time aiming at a flexible manufacturing system easy to deadlock. AB & B searches the branch tree of the system for the best solution to the problem. In the searching process, the AB & B selects nodes which are not branched under the guidance of the proposed branching strategy, and the branching strategy enables the AB & B to quickly update the optimal solution. The branching of the nodes that are not branched is done under the supervision of an existing optimal control strategy. In the searching process, pruning is carried out according to the lower bound of the node, and repeated searching is avoided by using a pruning rule. The branching strategy and pruning rules together enable the AB & B to quickly find the optimal or near optimal solution. The test results on the standard case show that AB & B is significantly superior to all existing methods, and the superior performance of AB & B is due to the proposed lower bounds, branching strategies and pruning rules.

In summary, the present invention has the following advantages:

(1) a lower bound and two pruning rules are provided to prune nodes without future and avoid repeated searching;

(2) a new branching strategy is provided, so that deadlock can be avoided, and the algorithm can be enabled to quickly update the current optimal solution;

(3) a fast branch-and-bound algorithm with only one parameter is provided, which is superior to all existing deadlock-free scheduling algorithms for flexible manufacturing systems.

Claims

1. A fast deadlock-free branch-and-bound scheduling method for a flexible manufacturing system is characterized by comprising the following steps:

step 1: flexible manufacturing system modeling

Step 1.1Petri Net preparation knowledge

a set of non-negative integers is represented,

a set of positive integers is represented that,

for directed arc sets, P and T are finite disjointIs not empty, i.e.

M′(p)＝M(p)-1；

m' (p) ═ M (p) + 1; otherwise, M' (p) ═ M (p); given a controller Δ, let M [ t, Δ > denote that initiating an enable transition t at an identification M is permitted by the controller Δ, enabling for a transition sequence τ t₁t₂...t_k，t_i∈T，i∈Z_kIf M is_i[t_i>M_i+1Wherein M is₁Let τ be feasible under M (feasible);

Wherein P ═ P₁∪P₂；T＝T₁∪T₂；F＝F₁∪F₂；

M₀(p)＝M₁₀(p)，

M₀(p)＝M₂₀(p)；

For the logo Petri Net (N, M)₀)＝(P,T,F,M₀) Let B (N, M)₀) Is (V, E) represents(N,M₀) Wherein V is (N, M)₀) E is a set of directed edges between nodes; a node consists of a transition sequence; let v₁And v₂Respectively indicates that two nodes satisfy M₀[v₁>M₁And M₀[v₂>M₂If T ∈ T is present, so that M₁[t>M₂Then there is a slave v₁To v₁Directed arc of, in a branch tree, node v₁Is of depth | v₁L +1, where v₁| represents v₁The number of medium transitions; all nodes at the same depth form a search layer, and the layer with the largest depth is called a bottom layer (bottom level);

And

given resource r_i∈ R, order

Indicates its initial resource amount, order

Wherein

Resource r_k∈ R is requested by multiple activities

Given a_ij∈A(r_k) Let path t_ijp_ijt_i(j+1)Representing its Petri Net model, where t_ijDenotes a_ijStart of) operation, t_i(j+1)Denotes a_ijTo finishIn operation, p_ijDenotes a_ijThe active position of (a); resource r_k∈ R resource location R_kIs represented by_kThe initial token number of C_kAdding two arcs, one from r_kTo t_ijRepresenting a request r_kThe other one from t_i(j+1)To r_kRepresents the release of r_kLet OPN_ijDenotes a_ijThe Petri Net model of (1);

for class u workpieces, two imaginary active positions p are added_u◇And p_u◆Respectively representing the start and end of the motion of the u-th workpiece, for theta_i∈Θ_uAdding two arcs, one from p_u◇To t_i1Another one from t_i(Li+1)To p_u◆The overall flexible manufacturing system is modeled as:

wherein P is a set of active positions;

M₀(p_u◇)＝Ψ_u；

M₀(p)＝0；

M₀(r_i)＝C_i；

For P ∈ P, let

Representing the processing time of the activity associated therewith, for

M_f(p_q◆)＝Ψ_q；

M_f(p)＝0；

step 2: deadlock-free fast branch-and-bound scheduling algorithm

step 2.1, lower bound of the node;

order to

Representing the first token or workpiece in p, assuming the workpiece

The time for entering p is tau (v)₁) Then, at time τ (v),

the remaining time in p is

Is the process path from p to p';

If M (p) is 0, let

Machining path

And is

If it is not

Then

Given r_ik∈Ξ_iLet I (r)_ikV) denotes that r is scheduled from τ (v) to the next activity_ikR between_ikA minimum idle time of; if r is_ikIs idle at τ (v), thatChinese character I (r)_ik,v)＝ξ(r_iV) - τ (v); otherwise

Wherein

Then

definition 1: given B (N, M)₀) Node v in (1), define

Theorem 1: given B (N, M)₀) Is connected to the node v in (1),

lower bound of completion time of v, i.e.

Step 2.2 defining branching strategies

And

are used to store branched and unbranched nodes respectively,

and

inputting: depth d of the current search layer;

initialization: d₁＝d，

flag＝true，v＝v₀Wherein v is₀Is an empty transition sequence;

step 2.2.1: when d is₁D ═ or

step 2.2.1.1: let d₁Is (d)₁+1)％d*；

step 2.2.2: if flag is true, then:

step 2.2.2.1: from

In selection

Minimum node v₁；

Step 2.2.2.2: v is to be₁From

Removing;

step 2.2.2.3: let M₀[v₁> M, let e (M) { T ∈ T | M [ T, PCF >);

step 2.2.2.4 for any change t ∈ E (M) { v₁t } adding to list;

step 2.2.2.5: let v equal v₁Where v is the selected unbranched node;

step 2.2.2.6: let d be (d)₁+1)％d*；

Step 2.2.3: returning flag, d, v and list;

step 2.3 definition of pruning rules

Once an unbranched node is selected and branched, it is inserted into

In the method, a new node generated by a branch is inserted into

And

in the process of (2):

pruning rule 1

Inputting: a branched node v;

initialization: let M₀[v＞M；d＝|v|+1；

Step A-1: if present

step A-1.1: if it is not

Then order

Is composed of

Finishing;

step A-2: order to

Is composed of

Pruning rule 2

Inputting: a newly generated unbranched node v;

initialization: let M₀[v＞M；d＝|v|+1；

Step B-1: if present

step B-1.1: if it is not

Then order

Is composed of

Finishing;

step B-2: if present

step B-2.1: if it is not

Then order

Is composed of

Order to

Is composed of

Finishing; step B-3: order to

Is composed of

Step 2.4, a deadlock-free rapid branching and delimiting algorithm;

Representing the current best solution completion time; given an unbranched node v, if

Then v is an unprivileged node because branch v cannot get a better solution than the current best solution, and the unprivileged node will be pruned during the search process;

at the beginning of the process, the process is carried out,