CN106651003A - Polynomial dynamic programming method of condition deflection approximate subgradient - Google Patents

Abstract

The invention discloses a polynomial dynamic programming method of a condition deflection approximate subgradient. The method comprises: implementing a machine capability Lagrangian relaxation strategy at smelting and refining stages, implementing a dynamic-programming-based relaxation problem approximation analysis method, solving a dual problem by using an approximate subgradient level algorithm, and implementing Constructing a list-scheduling-based heuristic type rule construction feasible solution algorithm. According to the machine capability Lagrangian relaxation strategy at smelting and refining stages, a relaxation strategy based on workpiece(melting) unit decomposition and a steel making-continuous casting rescheduling problem is solved by using the Lagrangian relaxation strategy. According to the construction method based on a workpiece unit constraint-relaxation strategy feasible solution, the basic idea is characterized in combining a workpiece starting processing time obtained by the relaxation problem, an objective function coefficient and a list scheduling method. And the dual problem of the steel making-continuous casting rescheduling problem is solved by using with approximate subgradient level algorithm with error controllability. Therefore, the production scheduling efficiency and quality can be improved obviously and the production rate can increase.

Description

A kind of condition deflects the multinomial dynamic programming method of approximate subgradient

Technical field

The invention belongs to areas of information technology, are related to operational research, optimized algorithm and steel smelting-continuous casting production programming, specifically relate to And a kind of condition deflects the multinomial dynamic programming method of approximate subgradient.

Background technology

Steelmaking-refining-continuous casting is the core link in Steel Production Flow Chart, including smelting, refining, the megastage of continuous casting three. In process of production, there is high requirement to molten steel processing and haulage time, liquid steel temperature and composition, need to meet production work On the premise of skill is required, the production capacity of equipment is given full play to, reduce material consumption and energy consumption, improve production efficiency.Actual production Cheng Zhong, molten steel processing and haulage time, the change of mechanical disorder can usually cause former operation plan to fail, and needs are resetted Degree, the solution of the problem has great significance to actual production.

The content of the invention

To solve the above problems, the invention provides a kind of condition deflects the multinomial Dynamic Programming side of approximate subgradient Method.

For achieving the above object, the technical scheme taken of the present invention is：

Condition deflects the multinomial dynamic programming method of approximate subgradient, including smelts the machine capability drawing with refining stage Ge Lang Relaxation Strategies, based on the relaxation problem approximate solution method of Dynamic Programming, approximate subgradient horizontal algorithm solve antithesis Problem and the heuristic rule based on list scheduling construct feasible resolving Algorithm；The smelting and the machine capability glug of refining stage Bright day Relaxation Strategy is solved using Lagrange Relaxation Strategies and refined using the Relaxation Strategy based on workpiece (heat) unit decomposition Steel-continuous casting weight scheduling problem；Specifically,

By introducing Lagrange multiplier { λ_i}

Loose constraint formula c_{I+1, S}-c_{I, S}-p_{I+1, S}>=0, i, i+1 ∈ Ω_k, 1≤k≤M_SAnd multiplier { λ (1)_{J, t}}

Loose constraint formula

Wherein,

Following relaxation problem is obtained

Wherein,

Constraints is formula

c_{I, s (i, g+1)}-c_{I, s (i, g)}-P_{I, s (i, g+1)}≥T_{S (i, g), s (i, g+1)}, i ∈ Ω, 1≤g ＜ S (7)

c_{I, S}-p_{I, S}+1≥A_{S, k}, i=b (k-1)+1,1≤k≤M_S (9)

c_{I, g}≥p_{I, s (i, g)}, i ∈ Ω, 1≤g≤S (10)

x_{I, g, t}∈ { 0,1 }, i ∈ Ω, 1≤g≤S, 1≤t≤T (11)

With multiplier constraint

λ_i>=0, b (k-1) ＜ i ＜ b (k), 1≤k≤M_S (14)

λ_{J, t}>=0,1≤j ＜ S, 1≤t≤T (15)

For given multiplier λ, relaxation problem (3) can be analyzed to the subproblem of the individual work cells of | Ω |, i.e.,

(16)

Wherein,

Wherein, the relaxation problem approximate solution method based on Dynamic Programming comprises the steps：

S1, from based on work cell decompose Relaxation Strategy：

Wherein, X is the feasible zone that not lax constraint is formed, L_i(λ, xⁱ) it is subproblem corresponding to workpiece i；Secondary In gradient algorithm, if L is (λ_k, x_k) for current iteration point functional value, λ_k+1For next iteration point, accurate relaxation problem is solved Method in, x_k+1Meet following condition

From formula (23), relaxation problem is made up of multiple subproblems, thus can consider to solve single subproblem, with full The sufficient tolerance that certain declines；

S2, set r ∈ (0, be 1) decreasing ratio, make s=0 for accumulative fall, i=1.

S3, order

S4, orderIf s >=rL is (λ_k+1, x_k), then make x_k+1=x_k, stop solving；Otherwise, i=i+1 is made, goes to next step；

If S5, i=be | Ω |, x is made_k+1=x_k, stop solving；If i is ＜ | Ω |, step S3 is gone to.

Wherein, the basic thought of the building method that Relaxation Strategy feasible solution is constrained based on work cell is with reference to lax Workpiece obtained by problem starts process time, objective function coefhcient and list scheduling method, specific as follows：

Step 1：If m_ijIt is processing machine sequence number (1≤ms of the workpiece i in operation j in initial schedule_ij≤M_j, 1≤j ＜ S), { c_ig| i ∈ Ω, 1≤g≤S } and { p_ij| i ∈ Ω, 1≤j≤S } for relaxation problem solution, T_jIt is set { t_{I, g}=c_{I, g}-p_ij | (i, g) ∈ O_jIn (1≤j≤S) the ascending arrangement of all elements form, T_j[n] is list T_jIn nth elements, J_{J, k} It is the set of the processing workpiece on the kth platform machine of j-th operation；

J=1, n=1 are made,|J_{J, k}|=0 (1≤k≤M_j)；

Step 2：Order

Step 3：If n≤| Ω |, goes to step 2；If j is ＜ S-1, j=j+1 is made, Go to step 2；Other situations, go to next step；

Step 4：If T_{J, k}It is set J_{J, k}The ascending arrangement of middle all elements forms, T_{J, k}[n] is list T_{J, k}In n-th Individual element, then make

Step 5：By in former problem machine capability constraint formula (1) replace with formula (25), while do not consider constraint formula (4)～ (6) and formula (11), using above- mentioned information, solving former problem will obtain a feasible solution of former problem.

Wherein, the antithesis for solving steel smelting-continuous casting weight scheduling problem using the controllable approximate subgradient horizontal algorithm of error is asked Topic, its algorithm is as follows；

Step 1, initialization：If initial value ε₁＞ 0, ε₁＞ ＞ ε₂＞ 0, λ₀>=0, δ₀＞ 0, β ∈ (0,1), t ∈ (0,1), σ_max＞ 0, N ＞ 0.Orderλ_best=λ₀, σ₀=0, k=0, r=0, l=0, s=0, M [l]=0, D [s]=0, h (l)=1/ (l+1), λ_best=λ₀, P_best=P (λ₀),

Step 2, functional value are calculated：

If step 2.1,Then makeλ_best=λ_k；Otherwise make

If step 2.2, P_best＜ P (λ_k), then make P_best=P (λ_k)；

Step 3, fully decline detection：IfThen make M [l+1]=k, σ_k=0, δ_l+1=δ_l, h (l+1)=h (l), l=l+1；

Step 4, weak underestimation detection：Make r=r+1,If r ＞ N and Then makeExpression is rounded downwards,

If r is ＞ N, r=0 is made；

Step 5, strong underestimation detection：If σ_k＞ h (l) σ_max, then M [l+1]=k is made,λ_k=λ_best, σ_k =0, δ_l+1=β δ_l, h (l)=1/ (l+1), l=l+1, s=s+1, D [s]=l；

Step 6, multiplier update：

Condition deflection approximate subgradient is defined as follows：

Wherein,It is abbreviated asIt is the condition approximate subgradient of F (λ), β_kFor deflection factor, T_Φ(λ) for Φ in λ The cone of tangents at place；

OrderMultiplier is updated by formula (26)～(28), wherein

Step 7, searching route add up：Order

Step 8, end condition：IfOrThen terminate iteration；Otherwise, step 2 is gone to.

The invention has the advantages that：

The efficiency and quality of production scheduling are significantly improved, productivity ratio is improve.

Description of the drawings

Fig. 1 is the workflow principle of the multinomial dynamic programming method that embodiment of the present invention condition deflects approximate subgradient Figure.

Specific embodiment

In order that objects and advantages of the present invention become more apparent, the present invention is carried out further with reference to embodiments Describe in detail.It should be appreciated that specific embodiment described herein is not used to limit this only to explain the present invention It is bright.

In following examples, the symbol definition for being used is shown in Table 1- tables 3.

Table 1 is indexed, element and set

The preset parameter of table 2

The decision variable parameter list of table 3

Based on time index variable mixed integer programming weight scheduling model：

Object function：MinG=G₁+G₂+G₃ (1)

Wherein,

In above formula, G₁For completion date and residence time penalty, G₂Time penalty function, G are poured for disconnected₃For stability Penalty.

For the nonlinear terms in cancelling (4), following two classes auxiliary variable is introduced：

Therefore,

Thus have

Constraint 1：Each operation of workpiece can only be processed on each operation machine,

Constraint 2：Each operation of each workpiece goes into operation the moment only once

Constraint 3：The completion date of workpiece has following relation with time index variable

Constraint 4：Heat machining sequence constraint, i.e., same heat in previous stage completion of processing and after transporting to next stage, Processing could be started,

c_{I, s (i, g+1)}-c_{I, s (i, g)}-P_{I, s (i, g+1)}≥T_{S (i, g), s (i, g+1)}, i ∈ Ω, 1≤g ＜ S. (11)

Constraint 5：In readjustment degree, the completion date of the operation for having gone into operation and process time are identical with former scheduling, i.e.,

Constraint 6：Any instant in readjustment degree, while the Number of Jobs that goes into operation is not waited more than its institute's energy process equipment Total number of units, this is machine capability constraint,

Wherein,

Constraint 7：The machining sequence constraint of the adjacent heat in same pour time, this also referred to as batch constraint,

c_{I+1, S}-c_{I, S}-p_{I+1, S}>=0, i, i+1 ∈ Ω_k, 1≤k≤M_S (14)

Constraint 8：The on-stream time of the constraint of final stage machine available time, i.e. workpiece can not be earlier than machinery equipment can With the time,

c_{I, S}-p_{I, S}+1≥A_{S, k}, i=b (k-1)+1,1≤k≤M_S (15)

Constraint 9：Variable bound,

c_{I, g}≥p_{I, s (i, g)}, i ∈ Ω, 1≤g≤S (16)

x_{I, g, t}∈ { 0,1 }, i ∈ Ω, 1≤g≤S, 1≤t≤T (17)

Embodiments provide the multinomial dynamic programming method that a kind of condition deflects approximate subgradient, including smelting With the machine capability Lagrange relaxation strategy of refining stage, based on the relaxation problem approximate solution method of Dynamic Programming, approximate Subgradient horizontal algorithm solves dual problem and the heuristic rule based on list scheduling constructs feasible resolving Algorithm；It is described smelting with The machine capability Lagrange relaxation strategy of refining stage is utilized using the Relaxation Strategy based on workpiece (heat) unit decomposition Lagrange Relaxation Strategies solve steel smelting-continuous casting weight scheduling problem；Specifically,

By introducing Lagrange multiplier { λ_iLoose constraint formula (13) and { λ_{J, t}Loose constraint formula (14)；It is obtained such as Lower relaxation problem

Wherein,

Constraints is the constraint of formula (8)～(12), formula (15)～(19) and multiplier

λ_j>=0, b (k-1) ＜ i ＜ b (k), 1≤k≤M_S (23)

λ_{J, t}>=0,1≤j ＜ S, 1≤t≤T (24)

For given multiplier λ, relaxation problem (20) can be analyzed to the subproblem of the individual work cells of | Ω |, i.e.,

Wherein,

Wherein, the relaxation problem approximate solution method based on Dynamic Programming comprises the steps：

S1, from based on work cell decompose Relaxation Strategy：

Wherein, X is the feasible zone that not lax constraint is formed, L_i(λ, xⁱ) it is subproblem corresponding to workpiece i；Secondary In gradient algorithm, if L is (λ_k, x_k) for current iteration point functional value, λ_k+1For next iteration point, accurate relaxation problem is solved Method in, x_k+1Meet following condition

From formula (32), relaxation problem is made up of multiple subproblems, thus can consider to solve single subproblem, with full The sufficient tolerance that certain declines；

S2, set r ∈ (0, be 1) decreasing ratio, make s=0 for accumulative fall, i=1.

S3, order

S4, orderIf s >=rL is (λ_k+1, x_k), then make x_k+1=x_k, stop solving；Otherwise, i=i+1 is made, goes to next step；

If S5, i=be | Ω |, x is made_k+1=x_k, stop solving；If i is ＜ | Ω |, step S3 is gone to.

Wherein, the basic thought of the building method that Relaxation Strategy feasible solution is constrained based on work cell is with reference to lax Workpiece obtained by problem starts process time, objective function coefhcient and list scheduling method, specific as follows：

Step 1：If m_ijIt is processing machine sequence number (1≤ms of the workpiece i in operation j in initial schedule_ij≤M_j, 1≤j ＜ S), { c_ig| i ∈ Ω, 1≤g≤S } and { p_ij| i ∈ Ω, 1≤j≤S } for relaxation problem solution, T_jIt is set { t_{I, g}=c_{I, g}-p_ij | (i, g) ∈ O_jIn (1≤j≤S) the ascending arrangement of all elements form, T_j[n] is list T_jIn nth elements, J_{J, k} It is the set of the processing workpiece on the kth platform machine of j-th operation；

J=1, n=1 are made,|J_{J, k}|=0 (1≤k≤M_j)；

Step 2：Order

Step 3：If n≤| Ω |, goes to step 2；If j is ＜ S-1, j=j+1 is made, Go to step 2；Other situations, go to next step；

Step 4：If T_{J, k}It is set J_{J, k}The ascending arrangement of middle all elements forms, T_{J, k}[n] is list T_{J, k}In n-th Individual element, then make

Step 5：Machine capability constraint formula (13) in former problem is replaced with into formula (34), while not considering constraint formula (8) ～(10) and formula (17), using above- mentioned information, solving former problem will obtain a feasible solution of former problem.

Wherein, the antithesis for solving steel smelting-continuous casting weight scheduling problem using the controllable approximate subgradient horizontal algorithm of error is asked Topic, its algorithm is as follows；

Step 1, initialization：If initial value ε₁＞ 0, ε₁＞ ＞ ε₂＞ 0, λ₀>=0, δ₀＞ 0, β ∈ (0,1), t ∈ (0,1), σ_max＞ 0, N ＞ 0.Orderλ_best=λ₀, σ₀=0, k=0, r=0, l=0, s=0, M [l]=0, D [s]=0, h (l)=1/ (l+1), λ_best=λ₀, P_best=P (λ₀),

Step 2, functional value are calculated：

If step 2.1,Then makeλ_best=λ_k；Otherwise make

If step 2.2, P_best＜ P (λ_k), then make P_best=P (λ_k)；

Step 3, fully decline detection：IfThen make M [l+1]=k, σ_k=0, δ_l+1=δ_l, h (l+1)=h (l), l=l+1；

Step 4, weak underestimation detection：Make r=r+1,If r ＞ N andThen makeRepresent to Under round,

If r is ＞ N, r=0 is made；

Step 5, strong underestimation detection：If σ_k＞ h (l) σ_max, then M [l+1]=k is made,λ_k=λ_best, σ_k =0, δ_l+1=β δ_l, h (l)=1/ (l+1), l=l+1, s=s+1, D [s]=l；

Step 6, multiplier update：

Condition deflection approximate subgradient is defined as follows：

Wherein,It is abbreviated asIt is the condition approximate subgradient of F (λ), β_kFor deflection factor, T_Φ(λ) for Φ in λ The cone of tangents at place；

OrderMultiplier is updated by formula (44)～(46), wherein

Step 7, searching route add up：Order

Step 8, end condition：IfOrThen terminate iteration；Otherwise, step 2 is gone to.

For ease of understanding above-mentioned subgradient horizontal algorithm, provide some and explain and explanation.

(1) step 3 is exactly that whole algorithm adjusts target state estimator level to step 5Strategy.Step 3 is that detection declines Whether amplitude is sufficiently large, if sufficiently large, current fall (δ is pressed always_l) be iterated；Step 4 is that detection algorithm is No have weak shock and swing (relative to step 5), if it is present directly adjusting the value of target state estimator level；Step 5 is detection algorithm Swing with the presence or absence of macroseism, i.e. target level valueIt is significantly underestimated, needs to be adjusted.It is worthy of note that step 3 and step 5 exist Play an important role in the analysis of Algorithm Convergence and convergency factor, in addition, step 5 adjustment target level valueMust be according to Certain rule, i.e., according to step 7 and the precondition and operation rule of step 5, cannot otherwise ensure convergence.

(2) M [l] and D [s] are the convenient symbols for carrying out theory analysis and explanation hereinafter and introducing, will not in Practical Calculation Use.What D [s+1]-D [s] was recorded is the iterations that algorithm fully declines, and M [l+1]-M [l] records are non-abundant decline Iterations.Whether W [r] records are history target function values, by the considerable functional values of measuring of W [r] a zonule Faintly shake, this concussion the reason for be to underestimate target function value, cause it to restrain, thus concussion.Here it is provided with The criterion epsilon of one concussion detection₂.Step 4 is primarily to avoid unnecessary redundancy iteration from improving efficiency with this, step 5 is then It is to estimate level value to ensure convergence by modification.

(3)δ₀Substantially F (λ)-F^*One estimation, σ_maxIt is then right | | λ₀-λ^*||₂One estimation.In Lagrange In relaxation method, δ₀Value be easier to obtain, F can be deduced by the object function of former problem^*Lower bound, and F (λ₀) as upper Boundary, has thus obtained F (λ)-F^*One estimation, can subsequently make σ_max=δ₀/||g₀||₂Obtain | | λ₀-λ^*||₂One it is thick Slightly estimate.It is worthy of note that, the estimate of above-mentioned parameter has no effect on convergence, simply during Practical Calculation The calculating time of possible minor way algorithm.In general, we set δ during Practical Calculation₀=τ (F (λ₀)+P(λ₀))(τ Why ∈ (0,1)), be added and be because F (λ₀)=- L (λ₀).According to the viewpoint of Kiwiel, above-mentioned subgradient horizontal algorithm can be with A simple version of beam subgradient method is regarded as, because each iteration of beam subgradient method needs directly calculating one secondary Planning problem is to obtain the subgradient direction of decline, and the step of said method 4 substantially instead of this direct meter to step 7 Calculation process, thus the method is more simple, while computation complexity is also much lower.

(4) above-mentioned subgradient horizontal algorithm is different from Goffin and Kiwiel methods maximum points out to be step 4 and increasing The adjustable strategies of the further optimal estimating level values of h (l), the two methods have been added to greatly improve the efficiency of algorithm, wherein h L the decrease speed of () cannot be below O (1/l).Further, since the method for Goffin and Kiwiel is directed to general convex optimization asking Topic, thus its end condition is whether subgradient is zero.But in Lagrange relaxation methods, subgradient is generally not equal to zero, But according to convergence proof, give such as the end condition in step 8, wherein δ_lEstimate equivalent to one of duality gap, For the history optimal value of dual function, when Lagrange relaxation methods solve the difficult integer programming problems of NP, duality gap with The ratio of dual function is generally higher than 10^-3, so end condition typically takes ε₁=10^-3。

Experimental result and analysis

Algorithm parameter is arranged and data instance

Condition-deflection approximate subgradient horizontal algorithm solves the calculation of the dual problem that Relaxation Strategy is constrained based on machine capability Method parameter setting is as follows：

ε₁=1e-3, ε₂=1e-5, t=0.8, W=4, β=0.8, δ₁=(P (λ₀)+F(λ₀))/5, σ_max=(P (λ₀)+F (λ₀))/||g_ε(λ₀) | |, λ₀=0.

The foundation that above-mentioned key parameter is arranged is explained：

(1)ε₁=1e-3 represents the amplitude proportional that dual function declines because the scheduling problem required by herein to be NP difficult Combinatorial optimization problem, thus there is duality gap, and also duality gap is generally higher than 1%, and this is summarized according to experiment law Arrive.If the duality gap for obtaining is less than 0.1% simultaneously, this is good enough in actual production, thus can terminate calculating Method.

(2)ε₂=1e-5 represents the 2- norm values of subgradient because dual problem be a Non-smooth surface continuous function, Thus if the norm value of subgradient is fully little, show current iteration point close enough dual problem optimum point, can be whole Only algorithm.

(3)δ₁=(P (λ₀)+F(λ₀))/5 substantially F (λ)-F^*One estimation because P (λ₀) it is the feasible of former problem Solution, consistently greater than dual problem optimal value, F (λ₀)=- L (λ₀) be dual problem a value, thus above-mentioned expression formula is F (λ)-F^*One estimation.

(4)σ_max=(P (λ₀)+F(λ₀))/||g(λ₀) | | it is then right | | λ₀-λ^*||₂One estimation.It is loose in Lagrange In relaxation method, δ₀Value be easier to obtain, F can be deduced by the object function of former problem^*Lower bound, and F (λ₀) as upper Boundary, has thus obtained F (λ)-F^*One estimation, can subsequently make σ_max=δ₀ ||g₀||₂Obtain | | λ₀-λ^*||₂One it is thick Slightly estimate.

(5)λ₀=0 represents that the usual default setting of initial multiplier is zero.

The foundation of the evaluation of Algorithm for Solving quality is duality gap, run time or iterations.Wherein duality gap Gap=(UB-LB)/LB × 100% is defined as, UB is best feasible solution (the as upper bound of problem), and LB is dual problem Solution (as problem lower bound).

The data of model are that the industrial real data based on Shanghai steel mill is equal at random in certain interval range Even generation, it is specific as follows：

(1) the operation sum S=4 of steel smelting-continuous casting production, number of units M of the parallel production equipment corresponding to each operation_j's Scope is 3≤M_j≤ 5, it is { 8,16,24,32 } that heat number is processed on every conticaster.

(2) refining procedure has two kinds of paths：RH → LF and LF → RH.The refining procedure path Shi Liangzhong roads of all batches Random uniform generation in footpath.

(3) transmission time T between adjacent operation_{J, j+1}Scope is 3≤T_{J, j+1}≤ 10 (units：Minute)；Standard adds man-hour Between P_ijScope is 36≤P_ij≤ 50, it allows the upper bound of adjustment and lower bound to be respectively 1.1P_ijAnd 0.9P_ij。

(4) penalty coefficient W₁=10+20 (S-1), W₂=10, W₃=30, W₄=20.

Event type

Consider following types of events：

(1) readjustment degree time point, it is considered to two time points：0.3C_maxAnd 0.7C_max, wherein C_maxFor the maximum of initial schedule Completion date.Note time point is 0.3C_maxEvent be R1, another is R2.

(2) process time delay, it is considered to which three class process times postponed event：It is standard zero, time delay that time delay is 0.1 times, 0.2 times that time delay is standard process time of process time.It is T1 that note delay is zero event, latter two difference For T2 and T3.

(3) machine-spoiled time, it is considered to three class machine-spoiled time events：Fault time is 0.03C_max, fault time be 0.06C_max.It is M1 that note fault time is zero event, and latter two is respectively M2 and M3.

Do not consider in text process time postpone and machine-spoiled time be zero situation, thus a total of 2 × 3 × 3-2 =16 kinds of event types, due to each categorical data example have it is random generate 10, therefore to consider 3 × 4 × 16 altogether × 10=1920 example.

For the ease of representing result, introduce some symbols and represent each algorithm and its problem：Number of stages represents with S, machine Number represents that batch number is represented with B with M, and the Number of Jobs in each batch is represented with J；

CS_Job：Represent based on work cell decomposition strategy and using the subgradient horizontal algorithm of condition subgradient.Accurately Method for solving bibliography.

CS_Job_A：Represent based on work cell decomposition strategy and using the approximate subgradient level of condition approximate subgradient Algorithm.Comparison of computational results and analysis based on the algorithm of approximate solution

By CS) based on Job_A algorithms, using different down ratio research approximate datas solution quality and solve effect Rate.Give under different down ratios (r) in different event lower aprons overall average duality gap, accurate overall average duality gap With the result of calculation of run time, wherein the lower bound in approximate overall average duality gap is the lower bound required by approximate data, and smart Really the lower bound in overall average duality gap is the lower bound required by exact algorithm, concrete outcome such as table 4- tables 6.Wherein, table 4 is algorithm The result of calculation of the approximate average duality gap of CS_Job_A, table 5 is the meter of the accurate average duality gap of algorithm CS_Job_A Result is calculated, table 6 is the average operating time of algorithm CS_Job_A.

The approximate overall average duality gap (%) of table 4CS_Job_A

Sequence number	Event type	R=0	R=0.02	R=0.04	R=0.06	R=0.08	R=0.1
								1	R1_T2_M1	-1.74	1.43	2.86	2.83	2.96	2.99
2	R1_T3-M1	-1.72	1.71	2.86	2.99	3.26	3.34
								3	R1_M1_M2	-1.69	0.99	2.30	2.30	2.11	2.03
4	R1_T2_M2	-1.67	1.50	2.76	2.80	2.87	2.91
								5	R1_T3_M2	-1.59	1.50	2.95	2.99	3.05	3.08
6	R1_T1_M3	-1.83	1.16	2.29	2.40	2.37	2.40
								7	R1_T2_M3	-1.65	1.37	2.70	2.75	2.77	2.79
8	R1_T3_M3	-1.50	1.60	2.91	2.99	3.05	3.08
								9	R2_T2_M1	-1.02	-8.03	-4.88	-5.65	-5.23	-4.97
10	R2_T3_M1	-0.74	-0.10	0.29	1.06	0.95	0.79
								11	R2_T1_M2	-1.13	-8.87	-7.57	-6.80	-5.63	-5.52
12	R2_T2_M2	-1.08	-6.96	-6.08	-7.39	-4.37	-3.37
								13	R2_T3_M2	-0.83	-4.39	-5.05	-4.29	-3.73	-3.36
14	R2_T1_M3	-1.17	-5.55	-5.72	-6.11	-5.43	-5.06
								15	R2_T2_M3	-1.04	-6.43	-7.20	-4.02	-1.86	-3.30
16	R2_T3_M3	-0.77	-1.57	-0.02	-1.98	-1.06	-1.01

The accurate overall average duality gap (%) of table 5CS_Job_A

Sequence number	Event type	R=0	R=0.02	R=0.04	R=0.06	R=0.08	R=0.1
								1	R1_T2_M1	3.09	2.77	2.89	2.86	2.99	3.02
2	R1_T3_M1	3.46	3.10	3.19	3.26	3.29	3.37
								3	R1_T1_M2	2.61	2.39	2.47	2.43	2.53	2.57
4	R1_T2_M2	2.99	2.76	2.79	2.83	2.90	2.94
								5	R1_T3_M2	3.22	2.96	2.99	3.03	3.09	3.12
6	R1_T1_M3	2.65	2.36	2.44	2.53	2.56	2.62
								7	R1_T2_M3	2.93	2.68	2.73	2.78	2.81	2.82
8	R1_T3_M3	3.19	2.93	2.95	3.03	3.09	3.11
								9	R2_T2_M1	0.99	0.91	0.91	0.91	0.92	0.91
10	R2_T3_M1	1.27	1.18	1.20	1.21	1.21	1.22
								11	R2_T1_M2	0.86	0.78	0.77	0.78	0.79	0.78
12	R2_T2_M2	0.96	0.87	0.85	0.87	0.88	0.87
								13	R2_T3_M2	1.15	1.05	1.06	1.06	1.05	1.06
14	R2_T1_M3	0.81	0.73	0.73	0.72	0.71	0.73
								15	R2_T2_M3	0.93	0.83	0.82	0.86	0.86	0.86

Overall average run time (s) of table 6CS_Job_A

Following observation is obtained by table 4- tables 6：

(1) for CS_Job_A, when down ratio r is not less than 0.04, the run time of algorithm and duality gap be not With essentially identical under down ratio.This it is meant that in practice down ratio more than after certain numerical value, it solves quality and solution Efficiency is not different.

(2) from table 1, the approximate duality gap of overall average of CS_Job_A is less than zero under many events, and this means Approximate lower bound is more than the upper bound.It is worthy of note that, most of minus events are the event containing R2.

Carried algorithm and actual production dispatching method comparison of computational results and analysis

Algorithm CS_Job_A is compared with actual production dispatching method.In order to compare solution effect, will be respectively compared Total complete time and (TC), total waiting time and (TS) obtained by each algorithm, total disconnected pour the time and (CB) refers to stability Mark (SM).The result of calculation of two kinds of algorithms such as following table.

Table 7CS_Job_A and actual production dispatching algorithm result of calculation

From the result of calculation of upper table 7, the solution quality of CS_Job_A is significantly better than actual production scheduling method, example As CS_Job_A total complete time and, total waiting time and, it is total disconnected pour the time and with stability indicator be respectively (30493, 13451st, 86,39), and the corresponding index of actual production dispatching method is respectively (49949,51808,1326,54).Carried algorithm It is that actual production dispatching method typically first fixes process time to be better than the reason for actual production scheduling is put, and then just determines to go into operation Time, and carried algorithm is while considering the two decision variables.It is above-mentioned test result indicate that, carrying algorithm can significantly improve The efficiency and quality of production scheduling, improves productivity ratio.

The above is only the preferred embodiment of the present invention, it is noted that for the ordinary skill people of the art For member, under the premise without departing from the principles of the invention, some improvements and modifications can also be made, these improvements and modifications also should It is considered as protection scope of the present invention.

Claims (4)

1. a kind of condition deflects the multinomial dynamic programming method of approximate subgradient, it is characterised in that including smelting and refining rank Machine capability Lagrange relaxation strategy, relaxation problem approximate solution method, the approximate subgradient water based on Dynamic Programming of section Flat Algorithm for Solving dual problem and the heuristic rule based on list scheduling construct feasible resolving Algorithm；The smelting and refining stage Machine capability Lagrange relaxation strategy using based on work cell decompose Relaxation Strategy, using Lagrange Relaxation Strategies Solve steel smelting-continuous casting weight scheduling problem；Specifically,

By introducing Lagrange multiplier { λ_i}

Loose constraint formula c_{I+1, S}-c_{I, S}-p_{I+1, S}>=0, i, i+1 ∈ Ω_k, 1≤k≤M_S (1)

With multiplier { λ_{J, t}}

Loose constraint formula

Wherein,

Following relaxation problem is obtained

$L\left(\mathrm{\λ}\right)=\min \{G+{\hat{G}}_1+{\hat{G}}_2\}---\left(3\right)$

Wherein,

Constraints is formula

$x_{i,g,t}=\underset{k=1}{\overset{M_j}{\&Sigma;}}{y}_{i,g,k,t},j=s(i,g),i\&Element;\mathrm{\&Omega;},1\&le;g<S,1\&le;t\&le;T---\left(4\right)$

$\underset{t=1}{\overset{T-P_{i,j}+1}{\&Sigma;}}{x}_{i,g,t}=1,j=s(i,g),i\&Element;\mathrm{\&Omega;},1\&le;g<S---\left(5\right)$

$c_{i,g}=\underset{t=1}{\overset{T-p_{i,j}+1}{\&Sigma;}}{\mathrm{tx}}_{i,g,t}+p_{i,j}-1,j=s(i,g),i\&Element;\mathrm{\&Omega;},1\&le;g<S---\left(6\right)$

c_{I, s (i, g+1)}-c_{I, s (i, g)}-P_{I, s (i, g+1)}≥T_{S (i, g), s (i, g+1)}, i ∈ Ω, 1≤g ＜ S (7)

$c_{i,g}={\stackrel{\&OverBar;}{c}}_{i,g},p_{ij}={\stackrel{\&OverBar;}{p}}_{ij},j=s(i,g),i\&Element;\mathrm{\&Omega;},1\&le;g<S---\left(8\right)$

c_{I, S}-p_{I, S}+1≥A_{S, k}, i=b (k-1)+1,1≤k≤M_S (9)

c_{I, g}≥p_{I, s (i, g)}, i ∈ Ω, 1≤g≤S (10)

x_{I, g, t}∈ { 0,1 }, i ∈ Ω, 1≤g≤S, 1≤t≤T (11)

$P_{i,j}^L\&le;p_{ij}\&le;P_{i,j}^U,i\&Element;\mathrm{\&Omega;},1\&le;j\&le;S---\left(12\right)$

$y_{i,g,k,t}^1\&Element;\{0,1\},y_{i,g,k,t}^2\&Element;\{0,1\},i\&Element;\mathrm{\&Omega;},1\&le;g\&le;S---\left(13\right)$

With multiplier constraint

λ_i>=0, b (k-1) ＜ i ＜ b (k), 1≤k≤M_S (14)

λ_{J, t}>=0,1≤j ＜ S, 1≤t≤T (15)

For given multiplier λ, relaxation problem (3) can be analyzed to the subproblem of the individual work cells of | Ω |, i.e.,

Wherein,

$f_{i,g}(\mathrm{\&lambda;},x,p,y)=w_{i,g}{c}_{i,g}+f_{i,g}^1+f_{i,g}^2+f_{i,g}^3---\left(18\right)$

$w_{i,g}=\left\{\begin{array}{cc}{W}_1+W_2-W_3-{\mathrm{\&lambda;}}_i,& i=b(k-1)+1,1\&le;k\&le;M_S,g=S;\\ W_1+W_2+W_3+{\mathrm{\&lambda;}}_i,& i=b\left(k\right),1\&le;k\&le;M_S,g=S;\\ W_1+W_2+{\mathrm{\&lambda;}}_i-{\mathrm{\&lambda;}}_{i-1},& b(k-1)+1<i<b\left(k\right),1\&le;k\&le;M_S,g=S;\\ -W_2,& i\&Element;\mathrm{\&Omega;},g=\left|O_i\right|+1;\\ 0,& i\&Element;\mathrm{\&Omega;},\left|O_i\right|+1<g<S;\end{array}\right.---\left(19\right)$

$f_{i,g}^1=\underset{t=1}{\overset{T-p_{ij}+1}{\&Sigma;}}\underset{\mathrm{\&tau;}=t}{\overset{t+p_{i,j}-1}{\&Sigma;}}{x}_{i,g,\mathrm{\&tau;}}{\mathrm{\&lambda;}}_{j,\mathrm{\&tau;}},j=s(i,g),1\&le;g<S---\left(20\right)$

$f_{i,g}^2=W_s\underset{t=1}{\overset{T}{\&Sigma;}}\underset{k=1}{\overset{M_{s(i,g)}}{\&Sigma;}}(y_{i,g,k,t}^1+y_{i,g,k,t}^2),1\&le;g<S---\left(21\right)$

$f_{i,g}^3=\left\{\begin{array}{c}({\mathrm{\&lambda;}}_i-W_3)p_{ij},b(k-1)+1<i\&le;b\left(k\right),1\&le;k\&le;M_S,g=S;\\ W_2{p}_{ij},i\&Element;\mathrm{\&Omega;},j=s(i,g),g=\left|O_i\right|+1;\\ -\underset{t=1}{\overset{T}{\&Sigma;}}\underset{k=1}{\overset{M_j}{\&Sigma;}}{\mathrm{\&lambda;}}_{j,t}\mathrm{\&delta;}(A_{j,k}-t),j=s(i,g),i\&Element;\mathrm{\&Omega;},1\&le;g<S.\end{array}\right.---\left(21\right).$

2. condition as claimed in claim 1 deflects the multinomial dynamic programming method of approximate subgradient, it is characterised in that described Comprised the steps based on the relaxation problem approximate solution method of Dynamic Programming：

S1, from based on work cell decompose Relaxation Strategy：

$L\left(\mathrm{\&lambda;}\right)=\underset{x\&Element;X}{min}L(\mathrm{\&lambda;},x)=\underset{x\&Element;X}{min}\underset{i=1}{\overset{\left|\mathrm{\&Omega;}\right|}{\&Sigma;}}{L}_i(\mathrm{\&lambda;},x^i)---\left(23\right)$

Wherein, X is the feasible zone that not lax constraint is formed, L_i(λ, xⁱ) it is subproblem corresponding to workpiece i；In subgradient In algorithm, if L is (λ_k, x_k) for current iteration point functional value, λ_k+1For next iteration point, in the accurate side for solving relaxation problem In method, x_k+1Meet following condition

$x_{k+1}=\arg \underset{x\&Element;X}{\min }L({\mathrm{\&lambda;}}_{k+1},x)---\left(24\right)$

From formula (23), relaxation problem is made up of multiple subproblems, thus can consider to solve single subproblem, to meet certain Plant the tolerance for declining；

S2, set r ∈ (0, be 1) decreasing ratio, make s=0 for accumulative fall, i=1.

S3, order

S4, orderIf s >=rL is (λ_k+1, x_k), then make x_k+1=x_k, stop solving；Otherwise, i=i+1 is made, goes to next step；

If S5, i=be | Ω |, x is made_k+1=x_k, stop solving；If i is ＜ | Ω |, step S3 is gone to.

3. condition as claimed in claim 1 deflects the multinomial dynamic programming method of approximate subgradient, it is characterised in that described The basic thought that the building method of Relaxation Strategy feasible solution is constrained based on work cell is to combine the workpiece obtained by relaxation problem Start process time, objective function coefhcient and list scheduling method, it is specific as follows：

Step 1：If m_ijIt is processing machine sequence number (1≤ms of the workpiece i in operation j in initial schedule_ij≤M_j, 1≤j ＜ S), { c_ig | i ∈ Ω, 1≤g≤S } and { p_ij| i ∈ Ω, 1≤j≤S } for relaxation problem solution, T_jIt is set { t_{I, g}=c_{I, g}-p_ij| (i, g) ∈O_jIn (1≤j≤S) the ascending arrangement of all elements form, T_j[n] is list T_jIn nth elements, J_{J, k}It is in jth The set of the processing workpiece on the kth platform machine of individual operation；

J=1, n=1 are made, |J_{J, k}|=0 (1≤k≤M_j)；

Step 2：Order $k^{*}=\arg \underset{1\&le;k\&le;M_j}{\min }{a}_{jk},J_{j,k^{*}}=J_{j,k^{*}}\&cup;\left\{i^{*}\right\},\left|j_{j,k^{*}}\right|=\left|J_{j,k^{*}}\right|+1,{\stackrel{\&OverBar;}{A}}_{j,k^{*}}={\stackrel{\&OverBar;}{A}}_{j,k^{*}}+p_{i^{*},j},m_{i^{*},j}=k^{*},n=n+1;$

Step 3：If n≤| Ω |, goes to step 2；If j is ＜ S-1, j=j+1 is made,Go to Step 2；Other situations, go to next step；

Step 4：If T_{J, k}It is set J_{J, k}The ascending arrangement of middle all elements forms, T_{J, k}[n] is list T_{J, k}In n-th yuan Element, then make

$c_{i_2,j}\&GreaterEqual;c_{i_1,j}+P_{i_2,j},i_2=T_{j,k}\&lsqb;n+1\&rsqb;,i_1=T_{j,k}\&lsqb;n\&rsqb;,1\&le;n<\left|J_{j,k}\right|,1\&le;k\&le;M_j---\left(25\right)$

Step 5：Machine capability constraint formula (1) in former problem is replaced with into formula (25), while not considering constraint formula (4)～(6) With formula (11), using above- mentioned information, solving former problem will obtain a feasible solution of former problem.

4. condition as claimed in claim 1 deflects the multinomial dynamic programming method of approximate subgradient, it is characterised in that utilize The controllable approximate subgradient horizontal algorithm of error solves the dual problem of steel smelting-continuous casting weight scheduling problem, and its algorithm is as follows；

Step 1, initialization：If initial value ε₁＞ 0, ε₁＞ ＞ ε₂＞ 0, λ₀>=0, δ₀＞ 0, β ∈ (0,1), t ∈ (0,1), σ_max＞ 0, N ＞ 0.OrderF (λ)=- L (λ), λ_best=λ₀, σ₀=0, k=0, r=0, l=0, s=0, M [l]=0, D [s]= 0, h (l)=1/ (l+1), λ_best=λ₀, P_best=P (λ₀),

Step 2, functional value are calculated：

If step 2.1,Then makeλ_best=λ_k；Otherwise make

If step 2.2, P_best＜ P (λ_k), then make P_best=P (λ_k)；

Step 3, fully decline detection：IfThen make M [l+1]=k, σ_k=0,δ_l+1= δ_l, h (l+1)=h (l), l=l+1；

Step 4, weak underestimation detection：Make r=r+1,If r ＞ N and Then make Expression is rounded downwards, K=k₁；

If r is ＞ N, r=0 is made；

Step 5, strong underestimation detection：If σ_k＞ h (l) σ_max, then M [l+1]=k is made,λ_k=λ_best, σ_k=0, δ_l+1=β δ_l, h (l)=1/ (l+1), l=l+1, s=s+1, D [s]=l；

Step 6, multiplier update：

Condition deflection approximate subgradient is defined as follows：

$d_{\mathrm{\&epsiv;}}^k={\hat{g}}_{\mathrm{\&epsiv;}}^k+{\mathrm{\&beta;}}_k{\hat{d}}_{\mathrm{\&epsiv;}}^{k-1}---\left(26\right)$

${\hat{d}}_{\mathrm{\&epsiv;}}^k=P_{-T_{\mathrm{\&Phi;}}\left(\mathrm{\&lambda;}\right)}\left(d_{\mathrm{\&epsiv;}}^k\right)---\left(27\right)$

${\mathrm{\&lambda;}}_{k+1}=P_{\mathrm{\&Phi;}}({\mathrm{\&lambda;}}_k-{\mathrm{\&alpha;}}_k{\hat{d}}_{\mathrm{\&epsiv;}}^k),k=0,1,2,\mathrm{...}---\left(28\right)$

Wherein,It is abbreviated asIt is the condition approximate subgradient of F (λ), β_kFor deflection factor, T_Φ(λ) it is Φ cutting at λ Cone；

OrderMultiplier is updated by formula (26)～(28), wherein

Step 7, searching route add up：Order

Step 8, end condition：IfOrThen terminate iteration；Otherwise, step 2 is gone to.