CN104850909B - Series-parallel system assignment problem optimization method based on Hungary Algorithm - Google Patents

Abstract

The invention discloses a kind of optimization methods of the series-parallel system task assignment based on Hungary Algorithm, belong to task scheduling technique field.System is divided into series connection link and link in parallel by the step of the present invention for (1)；(2) link in parallel is replaced with virtual work；(3) task assignment is carried out using traditional Hungary Algorithm；(4) whether judge the virtual work of assignment scheme can realize, change time arrow and return to step (3) if certain virtual work can not be realized, stop search if all virtual works can be realized and obtain final optimal scheduling scheme.The present invention has many advantages, such as that step is simple, search speed is fast and can obtain globally optimal solution, and can solve the task assignment problem of connection in series-parallel complication system, has expanded the application range of traditional Hungary Algorithm.

Description

Series-parallel system assignment problem optimization method based on Hungary Algorithm

Technical field

Task scheduling field of the present invention scope, specifically a kind of optimization method of assignment problem, can especially apply Into the research of the manufacture system task assignment problem with connection in series-parallel link.

Background technology

Assignment problem is the classical problem in personal scheduling problem --- m people completes n work, and everyone completes often The efficiency of item work is different, determines that task assignment scheme causes the total efficiency highest of completion task.Its universal model is as follows：

Wherein,

Since Hungary Algorithm has the characteristics that step is simple, can obtain optimal solution and need not verify, the algorithm is extensive In the solution of assignment problem, the extensive concern of domestic and foreign scholars is caused.

Classical Hungary Algorithm is that W.W.Kuhn utilizes Hungary mathematician D.Koning about neutral element independent in matrix The optimization method for being used to solve assignment problem that theorem proposes.The theoretical foundation of this method is：In beneficial matrix (also referred to as cost square Battle array) arbitrary row or column add or subtract a constant and will not change optimal distributing scheme^[11].Its basic thought is by every Row or each column add and subtract same constant to change beneficial matrix, until beneficial matrix different lines of not going together at least have that there are one null elements Element, and neutral element has just corresponded to the optimal distributing scheme of a total benefit minimum.The basic step of classical Hungary Algorithm is as follows：

Step 1, the beneficial matrix M for establishing resource allocation problem₀(m×n)。

Step 2, from beneficial matrix M₀Often row subtracts the element of row minimum so that often all there are one neutral elements for row, obtain M₁。

Step 3, from M₁Each column subtracts the element of row minimum so that all there are one neutral elements for each column, obtain M₂。

Step 4 covers M with minimum straight line₂In neutral element obtain M₃If the quantity of minimum straight line is equal to m, it is transferred to Step 6, otherwise it is transferred to step 5.

Step 5 matrix M₃In it is all not by straight line cover element subtract element minimum in uncovered element, simultaneously In straight line intersection point M is obtained plus the least member₄, enable M₂=M₄, go to step 4.

Step 6 is assigned since the minimum row or column of neutral element, is all assigned and is finished until all tasks, obtains best allotting Scheme P.

Although Hungary Algorithm has obtained larger application in practice, and achieves preferable effect.But it passes The Hungary Algorithm of system can only be directed to " total cost is the sum of each task cost " a kind of problem and be solved, and in Practical Project In, many situations are simultaneously unsatisfactory for this condition.Therefore, it is necessary to be improved to algorithm, so that it can preferably solve actually to ask Topic.

Invention content

For defect in the prior art, the present invention proposes that a kind of series-parallel system based on Hungary Algorithm is assigned Problem optimization method.

To solve problems of the prior art, technical solution proposed by the present invention is：

A kind of series-parallel system assignment problem optimization method based on Hungary Algorithm, it is characterised in that including following step Suddenly：

(1) system is divided into part in series and parallel connection part；

S (m, n)=C (a)+B (b)

In formula, S (m, n) represents whole system i.e. assignment problem, and m people completes n work；C (a) represents all series connection portions The work divided shares a；B (b) represents the work of parallel connection part, shares b link (B in parallel₁,B₂,…,B_b)；

(2) virtual work of parallel connection part and initialization are established；

Each parallel connection link is required for that there are one corresponding virtual worksTherefore the quantity of virtual work is also b, initially Change virtual workNamely determine the time arrow of virtual work

Wherein,Expression personnel i completes virtual workTime；

By virtual workInitial value be set as each personnel and complete virtual workTheoretical minimum timeRefer to that parallel operation may be completed most under the premise of personnel i participates in a job of link j in parallel The small time, i.e.,：

In formula, T_{ij_p}(p=1,2 ..., m_k) represent that personnel i completes the time that the work p in link j in parallel needs.m_kFor Working quantity in link j in parallel.

So, initial virtual, which works, is

(3) the new pure train S of the tandem working in original system and virtual work structure is utilized^V；

Corresponding time matrix

(4) using classical Hungary Algorithm to the pure train S of step (3)^VTask assignment is carried out, obtains allocation plan

Due to virtual work instead of link in parallel, assignment schemeIt must available free personnel R_free:

N_fFor idle personnel amount, and have：

(5) whether all virtual works can be achieved to judge allocation plan in judgment step (4)In all virtual works Make J^VWhether can realize, if it is go to step (7), otherwise turn (6)；

Virtual work J^VIt can be achieved to refer to by idle personnel R_freeAnd allocation planIn be assigned to completion this be virtual The link implementation in parallel of the composition of personnel of workTime meet：

Wherein T_pvkIt is schemeMiddle personnel p_vkComplete virtual workTime.Represent parallel connection link k in people Member p_vkAny one under participating in can realize deadline of scheme.

(6) change the efficiency vector of not achievable virtual work, return to step (3)；

A. if virtual work in scheme PMinimum time can be achieved and be less than the virtual work k corresponding timesI.e.：

WhereinRepresent parallel connection link k in personnel p_vkAny one under participating in is when can realize the completion of scheme Between.

So,Increase a unit, go to step 3.I.e.：

B. if virtual work in scheme PMinimum time can be achieved and be more than the virtual work corresponding time

ThenIt reduces by a unit, then goes to step 3.I.e.：

(7) virtual work is replaced with achievable parallel operation scheme, obtains optimal distributing scheme, optimization terminates.

Specifically, the virtual work in the present invention refers to all parameter attributes commonly to work, but is not specific work Make.

Virtual work quantity in step (2) is equal with parallel ring joint number amount, i.e., each link in parallel is one corresponding Virtual work.

The method have the benefit that：

The present invention improves traditional Hungary Algorithm, extends the application range of Hungary Algorithm, can Solve the task assignment problem of series-parallel system.The present invention is with step is simple, search speed is fast and can obtain globally optimal solution The advantages that.

Description of the drawings

Fig. 1 is the flow chart of the present invention

Fig. 2 is certain change system manufacturing flow chart

Fig. 3 is with the replaced system flow chart of virtual procedure

Specific embodiment

Optimization object model

Now there are one assignment problem S, by a tandem working (J₁,J₂,…,J_a) and b link composition (B in parallel₁,B₂,…, B_b), the working quantity of each parallel connection link is respectively m_i(i=1,2 ..., b), existing Num__RA personnel (R₁,R₂,…, R_{Num_R}), and：

The total time of a certain assignment scheme P is：

WhereinFor part in series work i need time,The time of work j for link i in parallel.

Known every personnel complete the time of work in every, it is desirable that everyone can only and must accomplish a task, problem It is that optimal assignment scheme P need to be looked for so that total time is minimum.

Related symbol defines

For convenience of explanation, the definition of related symbol is provided first.

S(m,n):Optimization object (i.e. the abstractdesription of assignment problem S), m people complete n work.

R(m):Personnel share m personnel.

R=[R₁,R₂,…,R_m]

Wherein R_i(i=1,2 ..., m) represent personnel i

J(n):Work shares n work

J=[J₁,J₂,…,J_n]

Wherein J_j(j=1,2 ..., n) represent work j

C(a):The part in series of S (m, n) shares a work, and has：

A ＜=n

B(b):The parallel connection part of S (m, n), shares b link in parallel, and each parallel connection link has b_iItem work.

B=[B₁,B₂,…,B_b]

Wherein B_kParallel connection part k in (k=1,2 ..., b) expression system.

J^Bi(q):Represent link B in parallel_iWork：

WhereinRepresent link B in parallel_iWork j.

T^S:Represent the time matrix (i.e. beneficial matrix) of S, the m * n matrix being made of the time of personnel's completion work, i.e.,：

Wherein T_ij(i=1,2 ..., m；J=1,2 ..., n) represent that personnel j completes the time of work i.

P^S：Represent a kind of assignment scheme of S

P^S=[p₁,p₂,…,p_n]

Specifically refer to personnel p_iComplete work i.

Time__P:The assignment scheme P of expression system S^STotal time：

Represent virtual work k

Represent the corresponding time arrows of virtual work k

WhereinExpression personnel i completes virtual workTime.

S^V(m,a+b)：Expression replaces the pure train that link in parallel obtains with virtual work, and m people, a+b work, Wherein there are b virtual works.I.e.：

Represent S^VTime matrix (beneficial matrix)

In formula, Ti (i=1,2 ... a) represent the time arrow of tandem working i,Represent virtual work j Time arrow.

Represent S^VA kind of assignment scheme

It is specific to represent personnel p_i, (i=1,2 ..., a) the work J of completion part in series_i；Personnel p_vi, (i=1,2 ..., b) Complete virtual work

T_P:Represent assignment schemeThe time of corresponding each work

Wherein, T_piExpression personnel p_i, (the work J of part in series a) is completed in i=1,2 ..._iTime, T_pviPersonnel p_vi, (i=1,2 ..., b) complete virtual workTime.

Represent the link B in parallel that virtual work k is represented_kA kind of implementation.

It is specific to represent personnel p_ki(i=1,2 ..., q) complete link B in parallel_kIn work

Expression schemeTotal time.

Wherein T_pkijExpression personnel p_kiComplete link B in parallel_kMiddle workTime.

It represents for S^VSchemeTotal time.

WhereinExpression personnel p_iComplete work J_iThe time needed,Complete virtual workThe time needed.

It is the flow chart of the present invention with reference to Fig. 1 for connection in series-parallel and the task assignment problem deposited, it is below in conjunction with the accompanying drawings, right Series-parallel system assignment problem optimization method the present invention is based on Hungary Algorithm is described in detail.

The detailed step of algorithm is as follows：

System is divided into part in series and parallel connection part by step 1：I.e.：

S (m, n)=C (a)+B (b)

In formula, S (m, n) represents whole system (assignment problem)；C (a) represents the work of all series connection, shares a；B(b) Represent link in parallel, quantity b.

The virtual work J of step 2 initialization parallel connection part B^V。

Each parallel connection link is required for that there are one corresponding virtual worksTherefore the quantity of virtual work is also b.Initially Change virtual workNamely determine the time arrow of virtual work

Theoretically, the time arrow of virtual work can be initialized as

It is 0 to assume that everyone completes time of virtual work, it is apparent that such initial point can cause it is a large amount of useless Iteration (because the working time of link in parallel is unlikely to be 0).

In order to improve the search efficiency of algorithm, herein by virtual workInitial value be set as each personnel complete it is empty Intend the theoretical minimum time of work k It refers under the premise of personnel i participates in a certain work of link j in parallel, The minimum time that parallel operation may be completed, i.e.,：

In formula, T_ijp(p=1,2 ..., m_k) represent that personnel i completes the time that the work p in link j in parallel needs.m_kFor simultaneously Join the working quantity in link j；

So, initial virtual, which works, is

Step 3 utilizes the new pure train S of the tandem working in original system and virtual work structure^V。

Corresponding time matrix

Step 4 is using traditional Hungary Algorithm to S^VTask assignment is carried out, obtains allocation plan

Although the number and number of tasks of former assignment problem are identical, due to, instead of link in parallel, being referred to virtual work Send schemeIt must available free personnel R_free:

N_fFor idle personnel amount, and have：

Step 5 judges allocation planIn all virtual work J^VWhether realize, if it is go to step 7, otherwise turn to walk Rapid 6.

Virtual work J^VIt can be achieved to refer to by idle personnel R_freeAnd allocation planIn completion virtual work people MemberThe personnel of virtual work, the link implementation in parallel of compositionTime meet：

Wherein T_pvkIt is schemeMiddle personnel p_vkComplete virtual workTime.Represent link k in parallel In personnel p_vkAny one under participating in can realize deadline of scheme.

Step 6 changes the efficiency vector of not achievable virtual work, return to step (3)；

A. if virtual work in scheme PMinimum time can be achieved and be less than the virtual work k corresponding timesI.e.：

WhereinRepresent parallel connection link k in personnel p_vkAny one under participating in is when can realize the completion of scheme Between.

So,Increase a unit, go to step 3.I.e.：

B. if virtual work in scheme PMinimum time can be achieved and be more than the virtual work corresponding time

ThenIt reduces by a unit, then goes to step 3.I.e.：

Virtual work is corresponded into time reduction herein and was referred to as excellent rollback principle, the purpose for setting excellent rollback principle is In order to avoid omitting optimal solution, theory deduction below can provide detailed description.

Step 7 replaces the virtual work in allocation plan P with achievable parallel operation scheme, obtains final most optimal sorting With scheme P^*, optimization terminates.

The step of in order to introduce improvement Hungary Algorithm proposed in this paper in more detail, and the validity of verification algorithm, this Text gives the example of certain equipment manufacturing system task scheduling.

Example

1st, problem describes

Certain equipment manufacturing system manufacturing process wherein there is 3 links in parallel, is remembered respectively as shown in Fig. 2, share 11 work P₁(J₂、J₃), P₂(J₅、J₆、J₇) and P₃(J₉, J₁₀).The system total process time is T_all：

Wherein T_Ji(i=1,2 ..., 11) represent work J_iThe time needed.

Existing 11 workers, and known every worker completes the time of each need of work, such as table 1.

1 worker of table completes the time that work in every needs

	J1	J2	J3	J4	J5	J6	J7	J8	J9	J10	J11
												R1	40	39	52	10	65	32	58	21	49	52	8
R2	41	33	49	15	68	30	60	26	48	56	12
												R3	39	35	47	14	55	38	54	24	42	52	15
R4	37	32	50	16	58	37	59	25	48	54	16
												R5	45	30	52	12	62	35	50	28	43	50	14
R6	36	34	48	18	57	34	52	24	45	48	18
												R7	38	34	49	13	60	36	53	26	45	46	16
R8	42	33	50	15	62	35	57	23	46	49	15
												R9	46	39	48	16	58	34	55	27	48	52	16
R10	38	38	52	14	56	38	59	24	47	54	13
												R11	39	36	50	19	64	34	54	29	46	48	17

In table, R_i(i=1,2 ..., 11) represents 11 workers, J respectively_i(i=1,2 ..., 11) represents 11 work respectively, Chronomere is minute.It is required that every worker has and completes a job, optimal task allocation plan is found.

2nd, the task distribution based on the method for the present invention

(1), it determines link in parallel, while calculates each personnel's minimum completion time of link virtual work in parallel.

Analysis chart 2 is P respectively it is found that the system totally 3 links in parallel₁(J₂、J₃), P₂(J₅、J₆、J₇) and P₃(J₉, J₁₀)。 Utilize 1 (J of virtual work_v1), 2 (J of virtual work_v2) and 3 (J of virtual work_v3) link in parallel is replaced, obtain new pure series connection System, such as Fig. 3.

Wherein,

Wherein T_{vi_j}(i=1,2,3；J=1,2 ..., 11) represent that worker j completes the time that virtual work i needs.

For boosting algorithm speed, the minimum virtual time that each worker of virtual work completes is calculated first.

T_{vi_j_min}=min (T_ik),k∈P_k

So as to obtain minimum time virtual work J_{v1_min}, J_{v2_min}And J_{v3_min}。

(2) minimum time virtual work (J is utilized_{v1_min}, J_{v2_min}, J_{v3_min}) and part in series work (J₁,J₄,J₈, J₁₁) structure time matrix M0:

(3) optimal distributing scheme of M0 is obtained according to traditional Hungary Algorithm.

Since M0 is not a square formation, according to document^[2]The step of " adding spot patch side " method, it is corresponding most that M0 can be obtained Excellent allocation plan.

Optimal case is as shown in table 2, and the scheme corresponding time is 182 minutes.

The corresponding optimal distributing schemes of table 2M0

(4) judge in the corresponding optimal distributing schemes of M0, the energy saving no realization of parallel ring.

It can be seen that, the corresponding optimal cases of M0 require R from table 2₅With R₄, R₉, R₁₀And R₁₁In 1 it is complete at 30 minutes Into link P in parallel₁Work (J₂、J₃)；It is required that R₂With R₄, R₉, R₁₀And R₁₁In 2 link P in parallel was completed at 30 minutes₂Work Make (J₅、J₆、J₇)；Require R simultaneously₃With R₄, R₉, R₁₀And R₁₁In 1 link P in parallel was completed at 42 minutes₃Work (J₉, J₁₀)。

Simple analysis can be found, by requirement above, link P in parallel₁The least cost time is 48 minutes, parallel ring Save P₂The least cost time is 58 minutes, link P in parallel₃The least cost time is 48 minutes.Therefore in the corresponding optimal cases of M0 3 virtual work (J_v1, J_v2, J_v3) it cannot all be converted into achievable link in parallel, that is to say, that optimal case at this time is not It can be achieved.

According to the step 6 of improved Hungary Algorithm, need the corresponding time (optimal case of M0 of two virtual works Middle virtual work J_v1, J_v2, J_v3Corresponding time, i.e. T_{v1_5},T_{v2_2},T_{v3_3}) improve a unit, i.e. T_{v1_5}：32 → 33, T_{v2_2}： 12 → 13, T_{v3_3}：12 → 13 obtain new virtual work.

Utilize J_v1, J_v2, J_v3And J₁, J₄, J₈, J₁₁Beneficial matrix M1 is built, return (3) continues search for feasible until obtaining Optimal solution.

3 optimum results and analysis

It is programmed and realized on the windowXp platforms that dominant frequency is 3.4GHz using MATLAB 2008a, by 226 iteration, 0.6s is taken, has finally obtained optimal dispatching scheme such as table 3, total time is 228 minutes, and the virtual work of iteration is at this time：

The final allocation optimum scheme of table 3

Using the virtual work in achievable link substitution tables 3 in parallel, complete optimal distributing scheme such as table 4 is obtained.

4 complete optimal distributing scheme of table

J1

J2

J3

J4

J5

J6

J7

J8

J9

J10

J11

R6

R4

R3

R5

R10

R2

R9

R8

R11

R7

R1

In order to verify whether the optimal solution that this algorithm obtains is global optimum, passes through MATLAB on identical platform herein Programming has traversed all possible scheme (totally 39916800), takes 59.7s, finds optimal allocation plan such as table 5, optimal Time is 228 minutes.

Optimal case in 5 traversing result of table

J1

J2

J3

J4

J5

J6

J7

J8

J9

J10

J11

R6

R4

R3

R5

R10

R2

R9

R8

R11

R7

R1

R6

R2

R3

R5

R10

R4

R9

R8

R11

R7

R1

Contrast table 4 and table 5, it is found that modified hydrothermal process of the present invention has obtained globally optimal solution really.

The above is only the preferred embodiment of the present invention, and protection scope of the present invention is not limited merely to above-mentioned implementation Example, all technical solutions belonged under thinking of the present invention all belong to the scope of protection of the present invention.It should propose, for the art Those of ordinary skill for, improvements and modifications without departing from the principles of the present invention, these improvements and modifications should also regard For protection scope of the present invention.

Claims (1)

1. a kind of series-parallel system assignment problem optimization method based on Hungary Algorithm, it is characterised in that include the following steps：

(1) system is divided into part in series and parallel connection part；

S (m, n)=C (a)+B (b)

In formula, S (m, n) represents whole system i.e. assignment problem, and m people completes n work；C (a) represents all part in series Work shares a；B (b) represents the work of parallel connection part, shares b link (B in parallel₁,B₂,…,B_b)；

(2) virtual work of parallel connection part and initialization are established；

Each parallel connection link corresponds to a virtual work J^V _k, therefore the quantity of virtual work is also b, initialization of virtual work J^V _k Namely determine the time arrow T of virtual work^V _k；

Wherein,Expression personnel i completes virtual work J^V _kTime；

By virtual work J^V _kInitial value be set as the theoretical minimum time T that each personnel complete virtual work k^V _{ij_min}；T^V _{ij_min} It refers under the premise of personnel i participates in a job of link j in parallel, the minimum time that parallel operation may be completed, i.e.,：

In formula, T_{ij_p}(p=1,2 ..., m_k) represent that personnel i completes the time that the work p in link j in parallel needs；m_kFor parallel connection Working quantity in link j；

So, initial virtual work is T^V _{k_min}：

(3) the new pure train S of the tandem working in original system and virtual work structure is utilized^V；

Corresponding time matrix

(4) using classical Hungary Algorithm to the pure train S of step (3)^VTask assignment is carried out, obtains allocation plan

Due to virtual work instead of link in parallel, assignment schemeIt must available free personnel R_free:

N_fFor idle personnel amount, and have：

(5) whether all virtual works can be achieved to judge allocation plan in judgment step (4)In all virtual work J^V Whether can realize, if it is go to step (7), otherwise turn (6)；

Virtual work J^VIt can be achieved to refer to by idle personnel R_freeAnd allocation planIn be assigned to and complete the virtual work The link implementation P in parallel of composition of personnel^V _kThe time of (k=1,2 ..., b) meets：

Wherein T_pvkIt is schemeMiddle personnel p_vkComplete virtual work J^V _kTime；Represent parallel connection link k in personnel p_vk Any one under participating in can realize deadline of scheme；

(6) change the time arrow of not achievable virtual work, return to step (3)；

A. if virtual work J in scheme P^V _kMinimum time can be achieved and be less than the corresponding time T of virtual work k^V _pvk, i.e.,：

WhereinRepresent parallel connection link k in personnel p_vkAny one under participating in can realize deadline of scheme；

So, T^V _pvkIncrease a unit, go to step 3；I.e.：

B. if virtual work J in scheme P^V _kMinimum time can be achieved and be more than the corresponding time T of virtual work^V _pvk,

Then T^V _pvkIt reduces by a unit, then goes to step 3；I.e.：

Virtual work is replaced with achievable parallel operation scheme, obtains optimal distributing scheme, optimization terminates.