CN103592920A - Hybrid flow shop scheduling method with finite buffers - Google Patents

Abstract

The invention relates to a hybrid flow shop scheduling method with finite buffers. Hybrid flow shop scheduling problems are typical production scheduling problems and belong to NP-hard problems. According to traditional scheduling methods, the buffers are supposed to be infinite mostly, but in actual production, buffers between adjacent working procedures are finite generally. A mixed integer programming model is established oriented to the hybrid flow shop scheduling problem with finite buffers, and an iterated variable neighborhood descent search algorithm is provided. In the algorithm, coding and decoding methods based on the sequence of workpieces are oriented to the characteristics of the hybrid flow shop scheduling problem. Large-scale neighborhood search based on block deletion and insertion is provided oriented to the problem that the wide-range search capacity of traditional neighborhood types is insufficient. The method can effectively ensure consistent material flow links between the working procedures, intermediate stock and energy consumption are reduced, and thus competitiveness of enterprises is improved.

Description

A kind of hybrid flow shop scheduling method that intermediate storage is limited in one's ability

Technical field

In manufacturing enterprise, the production of product all needs multiple working procedure just can complete conventionally, and every procedure generally has many identical processing machines.Therefore, this production scheduling problems can be summed up as hybrid flow shop scheduling problem (Hybrid Flowshop Scheduling Problem, HFSP).In traditional hybrid flow shop scheduling problem, conventionally suppose that the storage capacity between adjacent operation is unlimited.But, in actual industrial processes, between adjacent operation, be generally all provided with middle database, be used for storing intermediate product, with the continuity that guarantees that each operation is produced.Therefore, in actual schedule, also need the situation of considering that intermediate storage is limited in one's ability.

Background technology

The design of existing Flow Shop Scheduling solution, mostly can not consider the problem of storage capacity, yet in actual production process, interim stock inevitably exists.And to the scheduling problem solution in the limited situation of middle storage capacity, at present only for the design of Flow Shop Scheduling, and for the design of hybrid flow shop scheduling problem scheme still in blank.Therefore to the limited hybrid flow shop scheduling problem of middle storage capacity (HFSP with finite buffers, HFSP-FB) solution has great importance, and it can guarantee that inter process logistics is connected consistent, reduces interim stock, reduce energy consumption, thereby improve the competitive power of enterprise.

Summary of the invention

Technical matters to be solved by this invention is, on the basis of mixed-integer programming model of having set up intermediate storage hybrid flow shop scheduling problem limited in one's ability, based on becoming neighborhood decline search (iterated variable neighborhood descent search, VNDS) excellent properties that algorithm shows in the combinatorial optimization problems such as production scheduling, propose an iteration and become neighborhood decline search (iterated VNDS, IVNDS) algorithm.In this algorithm, the encoding and decoding method of the solution based on workpiece sequence has been proposed to reduce the difficulty of this problem neighborhood search, the neighborhood search method based on block delete and insertion of having proposed to be to improve the wide area search capability of algorithm, and in algorithm, introduced elite and collect to improve the ability that algorithm is jumped out local optimum.This model and algorithm does not all have specific (special) requirements process time with completing process time for prospect phase time of scheduling and the beginning of workpiece.

The technical solution used in the present invention is:

The mixed-integer programming model of 1.HFSP-FB problem

(1) problem is described

HFSP-FB problem as shown in Figure 1, comprises N workpiece, J continuous operation (stage), each stage j (j=1 ..., the number of identical parallel machines J) is Mj, has a middle database limited in one's ability between two adjacent stages.Each workpiece will be processed successively on a machine in each stage.When workpiece i is after the stage, j machined, if available free machine of next stage simultaneously this stage middle database below also has storage capacity, it both can directly be sent to the next stage by middle database and proceed processing so, also can be sent in this stage middle database below and store.When a machine m in certain stage is processing after a workpiece, if this stage middle database does not below have unnecessary storage capacity, this workpiece just must stay on machine m so, and machine m has just got clogged.Equally, if next stage does not have idle machine, certain the locational workpiece being stored in so in middle database also must stay on this position of middle database, and this position in middle database has got clogged.In addition, also require all workpiece all must before final delivery date, complete, do not allow to drag the phase.The target of problem is the weighted completion time that minimizes all workpiece.

(2) idea about modeling

In intermediate storage area between adjacent two stages, all comprise i memory location, and each position can be regarded the machine that a processing time is 0 as, therefore, each intermediate storage area is strongly fragrant can be regarded as one with the process segment of i parallel machine.Because these parallel machines are 0 to the processing time of workpiece, thereby can suppose that all workpiece all must process by all stages successively.After machining on the machine of a workpiece in certain stage, if the next stage does not have idle machine, this workpiece just must stay on this machine so, thereby this machine is got clogged.The time that machine gets clogged is not a constant, and it can utilize the time relevant with machine in the next stage the earliest.Like this, former problem does not have storage and workpiece without the HFSP problem of waiting for regard to having become a centre.This problem and former problem are of equal value.In problem after conversion, total number of stages becomes 2 * J-1, and wherein intermediate storage area is always in the even number stage.

(3) mixed-integer programming model

1. parameter

I work piece number, i ∈ 1,1={1,2 ..., n}, wherein n represents the sum of workpiece.

J stage No., j ∈ 1,2 ..., and S}, S represents total number of stages, S=2xJ-1, J is the total number of stages in former problem.

The machine number of k stage j.

The machine sum of Mj stage j.

The weight of wi workpiece i.

Pij workpiece i is in the processing time of stage j, notes at intermediate storage area stage j=2, and 4 ..., the pij=0 of 2xJ-2.

The final delivery date (dead1ine) of Di workpiece i.

Mono-of Q is not less than the number of scheduling prospect phase.

2. decision variable

If xijk workpiece i is dispensed on machine k and processes at stage j, xijk=1, otherwise xijk=0 (i ∈ 1:j=1 ..., S:k=1 ..., Mj).

If yilj comes at stage j workpiece i, workpiece 1 is tight to be processed above, yilj=1: otherwise yilj=0 (i, 1 ∈ 1:j=1 ..., S).

Cij workpiece i is in the deadline of stage j, i ∈ 1:j=1 ..., S.

Rij workpiece i is at the time departure of stage j, i ∈ I:j=1 ..., S.

The concrete meaning of parameter as defined above and decision variable as shown in Figure 2.

3. mixed-integer programming model

$\mathrm{Minimize}\underset{i=l}{\overset{N}{\mathrm{\Σ}}}{w}_i{c}_{\mathrm{iS}}---\left(1\right)$

$\begin{array}{c}S.t.\\ \underset{k=1}{\overset{N_1}{\mathrm{\Σ}}}{x}_{\mathrm{ijk}}=1,i\∈I;j=1,...,S---\left(2\right)\end{array}$

C _ij≥P _ij,i∈I

(3)

c _ij-P _ij≥c _i,j-1，i∈I；j=2，...，S (4)

c _ij-p _ij=d _i,j-1，i∈I；j＝2，...，S (5)

r _iS=C _iS，i∈I

(6)

c _iS≤D _i，i∈I

(7)

c _ij-p _ij+y _ijlQ≥r _ij，i,l∈I；j=2，...，S (8)

c _ij-P _ij+(1-Y _ijl)Q≥r _ij，i，l∈I；j=2，...，S (9)

c _ij-P _ij+(2+y _ijl-x _ijk-x _ijk)Q≥r _ij，i,l∈I；j＝2，...，S；k=1，...，M _j (10)

c _ij-p _ij+(3-y _ijl-x _ijk-x _ijk)Q≥r _ij，i,l∈I；j=2，...，S；k=1，...，M _j

(11)

c _ij≥0i∈I；j=1，...，S

(12)

r _ij≥0，i∈I；j=1，...，S

(13)

y _ijl∈{0，1}，i，l∈I

(14)

x _ijk∈{0，1}，i∈I；j=1，...，S；k=1，...，M _j (15)

Objective function (1) is the weighted completion time sum that minimizes all workpiece.Constraint (2) guarantees that each workpiece must be successively by all stages (comprising the intermediate storage stage), and can only be processed by a machine at each workpiece of each stage.Constraint (3) and (4) is the deadline constraint of workpiece, guarantees that each workpiece can not enter the next stage and start to process before the current generation does not machine.Constraint (5) shows that each workpiece is without waiting between the adjacent stage, starts processing in the next stage after leaving in the previous stage immediately.Constraint (6) illustrates that in the end a stage there will not be the obstruction of machine.Constraint (7) guarantees that each workpiece must complete processing before its final delivery date.Constraint (8)-(11) guarantee that every machine in each stage can not process two or more workpiece simultaneously, wherein constraint (8) and (9) is for certain stage, to only have the situation of a machine, and retrain (10) and (11), is for certain stage, to have the situation of many machines.The span of decision variable has been stipulated in constraint (12)-(15).

2. improved change neighborhood search algorithm

(1) encoding and decoding of separating

In order to reduce the complexity of neighborhood, do not use a complicated full schedule to be used as a solution of problem herein, but in the processing sequence of first stage, represent the coding of a solution with n workpiece, all neighborhoods move all and carry out for this coding.For a given coding, s=(s (1), s (2) ..., s (n)), wherein s (k) represents to come the work piece number of k position.Propose a structural formula algorithm herein and obtain this corresponding nearly excellent global solution of encoding.The principle (First Available Machine, FAM) of this algorithm based on utilizing the earliest machine, its concrete process can be described below:

Step1. put ri0=0 (i ∈ 1), and above the time of can utilizing the earliest of machine is 0 to put all stages.

Step2. put i=M1, front M1 workpiece in π is arranged into respectively on front M1 the machine in stage 1.

Step3. the time departure of each workpiece in calculation stages 1, and therefrom select time departure workpiece k (time departure of workpiece can utilize the maximal value of time the earliest for machine on deadline and the stage j+1 of this workpiece on stage j) the earliest, utilize FAM principle that workpiece k is arranged on the machine in all stages of residue.That upgrades upper machine of all stages can utilize the time the earliest, and calculates the beginning process time of workpiece k on each stage.

Step4. put i=i+1.If i≤n, is arranged into workpiece s (i) on the machine in stage 1 according to FAM principle, calculates the time departure of workpiece s (i) on the stage 1 and forward Step5 to: otherwise, stop, obtaining complete scheduling scheme.

Step5. identical with Step3, from the stage 1, select the workpiece complete the earliest, and according to FAM principle, be arranged on the machine in all stages of residue, and upgrade upper machine of all stages can utilize the time the earliest.Forward Step4 to.

This algorithm is all the workpiece of selecting first stage to complete the earliest at every turn, according to FAM principle, determines that it is in the machine assignment of Remaining Stages.Then the next workpiece to be processed in workpiece sequence is arranged on row first stage.Repeat this process until obtain a complete scheduling scheme.

In addition, because the target of problem is the weighted completion time sum that minimizes all workpiece, therefore we use a virtual processing time in this structural formula algorithm, be p ' ij=wj * pij, be used as the processing time of workpiece, thereby guarantee to consider deadline and the weight thereof of workpiece in the process of decoding.

(2) initial solution

Due to the sequence that is encoded to workpiece of separating, so we can adopt NEH method [16] to produce the initial solution of the HFSP-FB problem of studying herein.Its process is as follows:

Step1. calculate each workpiece i in the processing time in all stages sum, be designated as pi.Then according to the non-increasing order of relative value pi/wi all n workpiece sequencing.If the relative value of workpiece is identical, so just according to the ascending order at their final delivery date, sort.If this sequence be s=(s (1), s (2) ..., s (n)).

Step2. from this sequence, take out the first two workpiece s (1) and s (2), and suppose to only have these two workpiece to process, determine the optimal sequencing of these two workpiece, be assumed to be s '=(s ' (1), s ' (2)).

Step3. from workpiece sequence s, take out next workpiece, and it is inserted in the desired positions of resulting solution s ' above to (make the recruitment of objective function minimum).Repeat this step until all workpiece are all scheduled, obtain all workpiece till the sequence (initial solution) of first stage.

(3) neighborhood search adopting

For take the production scheduling problems of workpiece sequence part as the coding of solution, traditional neighborhood type is mainly to move with insertion and move based on swap.It is two adjacent or non-conterminous workpiece that exchange at random in workpiece sequence that Swap moves, and insertion moves first workpiece of random erasure from a workpiece sequence, and then by its radom insertion to other position.These two kinds of neighborhood types comparatively simply and easily realize, and therefore, many algorithms for production scheduling problems all adopt this two kinds of neighborhood types [17]-[20].But the scale of these two kinds of traditional neighborhood types is less, its complexity is respectively O (n (n-1)/2)=O (n2) and O (n (n-1))=O (n2), and its wide area search capability need to improve.Therefore, proposed the extensive neighborhood search based on block delete and insertion herein, this neighborhood can taken into account under the prerequisite of local ability, improves algorithm and jumps out the ability of local optimum, thereby increase its wide area search capability.

The insertion of this neighbour structure based on traditional moves, but to move be to carry out for a series of continuously arranged workpiece pieces for this.Suppose that this workpiece piece consists of adjacent 1 (1 >=1) individual workpiece, so for a given solution sb=(s (1) ..., s (n)), the local search procedure based on this neighbour structure is as follows:

Step1., iterations k=1 is set, and sb=s is preferably separated in order.

Step2. select at random 1 adjacent workpiece in s, and deleted, suppose that the order of deleted workpiece is designated as s (d1), s (d2) ..., s (d1), the partial solution that residue workpiece forms is designated as s '.

Step3., j=1 is set.

Step4. s (dj) is inserted into (after being inserted on this position, the recruitment of objective function is minimum) on position best in partial solution s '.

Step5., j=j+1 is set.If j≤1, forwards Step4 to: otherwise stop, a new global solution s ' obtained.

Step6., j=j+1 is set.If j≤1, forwards Step4 to: otherwise, obtain a new global solution s ', forward Step7 to.

If Step7. separate s ', be better than separating sb, make s=sb=s '.

Step6., k=k+1 is set.If k≤kmax (maximum iteration time), forwards Step2 to: otherwise stop output sb.

The complexity of above-described new neighborhood search is O (kmaxln), and when kmax=n, its complexity is O (ln2).Because the movement based on block delete and insertion can make algorithm, in larger solution space, search for, thereby can strengthen the ability that neighborhood search algorithm is jumped out local optimum region.In the VNS algorithm proposing at us, all neighborhood types are the piece neighborhood described in this section.Meanwhile, also introduced the thought that becomes deep search, made ascending variation of size of neighborhood, the Size of Neighborhood in VNS, from 1=1, increases to 1=L (total L neighborhood type) gradually.For the purpose of simple, this L neighborhood type is designated as to N1 successively, N2 ..., NL.

(4) iteration VNDS algorithm flow

As previously mentioned, in order to strengthen algorithm, jump out the ability of local optimum, in VNS algorithm, also introduced elite and collected E, be used for storing the resulting front b of VNS algorithm and preferably separate.In each iterative process of iteration VNDS algorithm, initial solution is chosen at random from this set E.

The flow process of the iteration VNDS algorithm proposing can be described down:

Step1., iterations g=1 is set, maximum iteration time gmax=100, maximum running time T max=600 second.Each neighborhood type of setup and use is carried out the maximum iteration time kmax=5 in Local Search, the big or small b=10 of set E.

Step2. use 3.The described NEH algorithm of 2 joint obtains an initial solution, and deposits it in set E.

Step3. from set E, choose at random one and separate s.

Step4. use N1 neighborhood type (being 1=1) to carry out Local Search to separating s, the preferably solution obtaining is designated as s ' 1.If s ' 1 is better than gathering the poorest solution in E, s ' 1 is inserted in set E: if the quantity of separating in set E after inserting surpasses b, the poorest solution in set E is deleted.

Step5. use N2 neighborhood type (being 1=2) to carry out Local Search to separating s ' 1, the preferably solution obtaining is designated as s ' 2.If s ' 2 is better than gathering the poorest solution in E, s ' 2 is inserted in set E: if the quantity of separating in set E after insertion surpasses b, by the most also separating and deleting in set E.

Step6. use N3 neighborhood type (being 1=3) to carry out Local Search to separating s ' 2, the preferably solution obtaining is designated as s ' 3.If s ' 3 is better than gathering the poorest solution in E, s ' 3 is inserted in set E: if the quantity of separating in set E after inserting surpasses b, the poorest solution in set E is deleted.

Step7. use N4 neighborhood type (being 1=4) to carry out Local Search to separating s ' 3, the preferably solution obtaining is designated as s ' 4.If s ' 4 is better than gathering the poorest solution in E, s ' 4 is inserted in set E: if the quantity of separating in set E after inserting surpasses b, the poorest solution in set E is deleted.

Step8., g=g+1 is set.If g>gmax or current computing time surpass Tmax, algorithm stops, the final solution that the preferably solution in output set E arrives as algorithm search: otherwise, turn Step3.

According to above description, can find out, iteration VNDS algorithm, in each iterative process, has all carried out a VNDS search, and the scale of neighborhood becomes greatly gradually, and the search depth of algorithm increases gradually, and the wide area search capability of algorithm also progressively strengthens simultaneously.

Accompanying drawing explanation

The hybrid flow shop scheduling problem schematic diagram that Fig. 1 intermediate storage is limited in one's ability;

The definition schematic diagram of Fig. 2 correlation parameter and decision variable.

Claims (4)

1. an intermediate storage hybrid flow shop scheduling method limited in one's ability, is characterized in that: the encoding and decoding method that has designed the solution based on workpiece processing sequence.

2. a kind of intermediate storage claimed in claim 1 hybrid flow shop scheduling method limited in one's ability, is further characterized in that: use the extensive neighborhood search based on block delete and insertion.

3. a kind of intermediate storage claimed in claim 1 hybrid flow shop scheduling method limited in one's ability; be further characterized in that: on the basis of claim 2; the thought that has added degree of deepening; the neighborhood scale based on block delete and insertion can change from small to large along with the carrying out of search, and the center of gravity of search is transformed into wide area search from Local Search gradually.

4. a kind of intermediate storage claimed in claim 1 hybrid flow shop scheduling method limited in one's ability, be further characterized in that: on the basis of claim 3, in becoming neighborhood decline search, introduced elite's collection, the better solutions obtaining in memory search process, the initial solution that becomes the each iteration of neighborhood search is concentrated and is chosen at random from this elite.

Images