Online scheduling method considering periodic maintenance of batch machines

Abstract

The application provides an online scheduling method considering periodic maintenance of a batch processing machine, taking the batch processing machine as a research object, considering the batch processing machine in the periodic maintenance process, and solving the actual problems of online scheduling of orders arriving over time, batch grouping, processing and the like. For various common scheduling objectives, such as maximum completion time of a workpiece, maximum flow time of a workpiece in a production system and the like, an effective online algorithm is designed through a waiting strategy to generate a real-time scheduling scheme, and the best value that can be reached by the algorithm performance under various objectives is analyzed, so that the online scheduling rules and methods for solving the online scheduling problem of the batch processing machine under the periodic maintenance are provided, guidance is provided for production planning of an enterprise, and a referable technical means is provided for other batch processing scheduling problems of the enterprise.

Description

TECHNICAL FIELD

[0001] The present application relates to the technical field of batch processing machine online scheduling, in particular to an online scheduling method considering periodic maintenance of batch processing machine. BACKGROUND

[0002] Batch processing machine can process multiple workpieces at the same time, improving the processing capacity of the machine and breaking through the limitation that the machine can only process one workpiece at a time. At the same time, the batch processing machine has high initial investment cost, and its running situation greatly affects the input-output capacity of the production workshop.

[0003] High-temperature heat treatment machine is a typical batch processing machine, and the processing process requires that the processed forgings (hereinafter referred to as workpieces) be put into the furnace in batches to increase the processing efficiency. Generally, the high-temperature heat treatment machine cannot run continuously for a long time, otherwise it will cause downtime. In order to avoid downtime, it needs to be maintained regularly to keep it running normally.

[0004] In order-oriented production systems, workpieces are processed according to orders, and the specific information of the order workpieces is unknown before the order is placed, which is called online scheduling in academic research. The traditional plan-based production method is difficult to cope with changes in temporary demand, and online real-time decision-making is needed in the short term. In order to ensure the orderly, smooth, balanced and efficient operation of such batch processing machines in the production process, efficient online rules or algorithms need to be designed. However, the demand for periodic maintenance of batch processing machines has uncertainty, which increases the difficulty of designing scheduling algorithms and performance analysis.

[0005] Existing scheduling research and technology on batch processing machines include offline and online environments. However, in offline research, the scheduling problem of batch processing machines does not involve machine maintenance. Through retrieval, it is found that there is currently no academic or technical research precedent on the online scheduling problem of batch processing machines involving machine maintenance. SUMMARY

[0006] To solve the problems in the prior art, the present application realizes real-time online scheduling of workpieces by periodic maintenance of batch processing machines, minimizes scheduling targets, and takes the batch processing machine as the research object, considers the online scheduling of orders arriving over time during the periodic maintenance of the batch processing machine, and solves the actual problems of batch grouping and processing.

[0007] The technical scheme of the present application is as follows:

[0008] The online scheduling method considering periodic maintenance of the batch processing machine comprises the following steps:

[0009] Step 1: mathematical modeling of the online scheduling problem of the batch processing machine under periodic maintenance:

[0010] The established model is as follows:

[0011] Order workpiece J j arrives over time, and the information of the workpiece is unknown when it does not arrive;

[0012] The workpiece needs to be processed on m batch processing machines, and the machine set is denoted as {M1, M2, …, M m};

[0013] The batch processing machine can process b workpieces at the same time, and the processing time of each batch of workpieces is fixed as 0<p<1;

[0014] All machines need to be periodically maintained regularly, and the maintenance time length is T; during the maintenance process, the machine cannot process workpieces;

[0015] The available time of the machine between two maintenances is set as a unit time 1, and the available time interval of the machine is denoted as [l k ,l k +1], wherein l1=0, and l k+1 =l k +1+T, k∈N + ;

[0016] Each workpiece needs to be immediately transported to the customer after the completion time C j , and the transportation time is denoted as q j , and the vehicle is sufficient; only when the workpiece reaches the customer, the workpiece flow is considered to be completed;

[0017] Optimization scheduling objective is to minimize D max = max j {C j + q j}, DF max = max j {C j + q j - r j} ; wherein the arrival time of the workpiece J j is r j , the completion time C j , the transportation time is recorded as q j ; D max is the maximum transportation completion time, and DF max is the maximum process time.

[0018] Step 2: Obtain the current time t, the number of machines m, the batch capacity b, the batch processing time p, and the information of the workpieces that have arrived at the current time, including the arrival time and the transportation time;

[0019] Step 3: Process each available time interval as follows:

[0020] Let [l, l+1] be any available time interval;

[0021] Set x, y, satisfying xp+yp=1, wherein 0≤x<1, y≥1 is a positive integer;

[0022] Set the sub-interval as:

[0023]

[0024]

[0025] Step 3.1: When the batch capacity is sufficient, use the online algorithm Infinite Capacity Delay (ICD) to solve;

[0026] The online algorithm ICD is to set the LT time in each interval: Except for the first interval, each sub-interval with a length of p has m LT times; if the current time t is a certain LT time and there is a machine with free time, then all workpieces are processed on the machine M i+1 at this time; otherwise, wait until the next LT time;

[0027] Step 3.2: When the batch capacity is limited and the optimization objective is D max , apply the Full Processing (FP) algorithm to solve;

[0028] The FP algorithm is: if the current time t is the starting time of a sub-interval with length p , and there is a machine idle at this time, then all jobs are grouped into full batches as much as possible according to the minimum release time first rule; suppose that a batches are formed, and the first a-1 batches are full batches; then the first min{m, a} batches are placed at time on any idle machine;

[0029] Step 3.3: in the case of limited batch capacity and the optimization target being DF max , the Full Processing under Delivery (FPD) algorithm is applied for solving;

[0030] The FPD algorithm is: if the current time t is the starting time of a sub-interval with length p , and there is a machine idle at this time, then all jobs are grouped into full batches as much as possible according to the D-ERD rule; suppose that a batches are formed, and the first a-1 batches are full batches; then the first min{m, a} batches are placed at time on any idle machine; the D-ERD rule is: jobs are sorted according to their q j -r j values in non-decreasing order, and if there are multiple jobs with the same q j -r j value, then the jobs are further sorted according to the ERD rule;

[0031] Step 4: at time t, it is determined whether to perform batch processing using the algorithm in Step 3; if time t satisfies the processing condition, then the scheduling scheme is output at the current time t according to the above algorithm and processing is performed.

[0032] Further, in Step 3, for the maximum completion time D max , for the case of sufficient batch capacity, there is no online algorithm with a competitive ratio less than , and when m>1, there is no online algorithm with a competitive ratio less than ; for the case of limited batch capacity, there is no online algorithm with a competitive ratio less than min{2, 1+α}; α is the positive root of α 3 +2α 2 -(1+T)=1+T; for the maximum flow time DF max , for the case of sufficient batch capacity, there is no online algorithm with a competitive ratio less than 1+1 / m; for the case of limited batch capacity, there is no online algorithm with a competitive ratio less than 2.

[0033] Further, in Step 3, for the case of sufficient batch capacity, the targets D max and DF maxThe competition ratio of the online algorithm ICD is 1+1 / m.

[0034] Further, in step 3, for the case of limited machine batch capacity, and the target is D max The competition ratio of the online algorithm FP is 2.

[0035] Further, in step 3, for the case of limited machine batch capacity, and the target is DF max The competition ratio of the online algorithm FPD is 2.

[0036] Advantages

[0037] The online scheduling method considering periodic maintenance of a batch machine provided by the application can generate real-time scheduling schemes through a waiting strategy design effective online algorithm for various common scheduling targets, such as maximum completion time of a workpiece and maximum flow time of a workpiece in a production system, and the algorithm performance can reach the best value under the corresponding target.

[0038] Additional aspects and advantages of the application will be set forth in part in the description which follows, and in part will become apparent to those skilled in the art upon examination of the following and / or can be learned by practice of the application. BRIEF DESCRIPTION OF DRAWINGS

[0039] The above and / or additional aspects and advantages of the application will become apparent and be readily appreciated from the description of the embodiments, taken in conjunction with the following drawings in which:

[0040] Figure 1 : sub-interval of the available time interval;

[0041] Figure 2 : flowchart of the application;

[0042] Figure 3 : sub-interval of the available time interval [0, 1] of example 1;

[0043] Figure 4 : online ICD algorithm scheduling scheme;

[0044] Figure 5 : online FP algorithm scheduling scheme. DETAILED DESCRIPTION

[0045] The application is directed to the online scheduling optimization problem of a workpiece in a periodic maintenance environment of a batch machine, proposes an online scheduling model in the periodic maintenance process of a batch machine, considers various common scheduling targets, determines the online scheduling scheme of a batch machine processing a workpiece through algorithm design, and analyzes the best performance that can be reached by the algorithm under various targets.

[0046] The basic steps are: firstly, constructing a mathematical model of the problem; then collecting information of the number of machines and other processing parameters; then designing and using online algorithm rules; and finally outputting the processing scheduling scheme at the time of deciding to process the batch.

[0047] Step 1: Mathematical modeling of the online scheduling problem of batch machines under periodic maintenance:

[0048] The online scheduling problem of batch machines under periodic maintenance can be established as:

[0049] Order workpiece J j arrives over time, and its information is unknown when it has not arrived. These workpieces need to be processed on m batch machines, and the machine set is denoted as {M1, M2, …, M m}. A batch machine can process b workpieces at the same time, and the processing time of each batch of workpieces is fixed at p < 1. All machines need periodic regular maintenance, and the maintenance time length is T. During the maintenance process, the machine cannot process workpieces.

[0050] It is assumed that the available time of the machine between two maintenances is a unit time 1, so the machine available time interval is denoted as [l k , l k +1], where l1=0, and l k+1 =l k +1+T, k∈N + . Each workpiece needs to be immediately transported to the customer after the completion time C j , and the transportation time is denoted as q j , and the vehicle is sufficient. Only when the workpiece reaches the customer, the workpiece flow is considered to be completed. The scheduling objective is to minimize D max =max j {C j +q j}, DF max =max j {C j +q j -r j}. It is assumed that the arrival time of the workpiece J j is r j , the completion time C j , and the transportation time is denoted as q j . D max is the maximum transportation completion time, and DF max is the maximum flow time. The present invention only involves the above two optimization objectives.

[0051] Step 2: Input the current time t, the number of machines m, the machine batch capacity b, and the batch processing time p. Input the information of the workpieces that have arrived and have not been processed at the current time, including their arrival time and transportation time.

[0052] Step 3: Constructing online algorithms. Process each available time interval.

[0053] Let [l, l+1] be any available time interval;

[0054] Set x, y such that xp+yp=1, where 0≤x<1, y≥1 is a positive integer;

[0055] Set sub-interval As shown in Table 1 and Figure 1 ;

[0056] Table 1 Sub-interval setting

[0057]

[0058] Step 3.1: When the batch capacity is sufficient, and the optimization goal is D max or DF max , apply the online algorithm Infinite Capacity Delay (ICD), which is to delay each workpiece at most (1 / m)p time to start in the sub-interval.

[0059] Algorithm ICD: Set Ladder Time (LT) in each interval: Except for the first interval, each sub-interval of length p has m LT times. If the current time t is a certain LT time and there is a machine idle (here u<y), then all workpieces at this time are formed into a batch on machine M i+1 . Otherwise, wait until the next LT time.

[0060] Step 3.2 When the batch capacity is limited, and the optimization goal is D max , apply the Full Processing (FP) algorithm.

[0061] Algorithm FP: If the current time t is the start time of a certain sub-interval of length p and there is a machine idle at this time, then all workpieces are formed into full batches as much as possible according to the Earliest Release Date First rule (ERD). Assume that a batches are formed, and the first a-1 batches are full batches. Place the first min{m, a} batches at time on any idle machine.

[0062] Step 3.3 In the case of limited batch capacity, and the optimization goal is DF max , apply the Full Processing under Delivery (FPD) algorithm. Define the D-ERD rule: workpieces are processed according to their qj -r j Values are sorted in non-decreasing order, and if there are multiple jobs with the same value, then they are sorted according to the ERD rule.

[0063] Algorithm FPD: If the current time t is the starting time of a sub-interval of length p and the machine is idle at this time, then all jobs are grouped into full batches as much as possible according to the D-ERD rule. Suppose that a batches are formed, and the first a-1 batches are full batches. The first min{m, a} batches are processed on any idle machine at time .

[0064] Step 4: At time t, it is determined whether to process batches using the above algorithm. If the time t satisfies the processing condition, then the scheduling scheme is output at the current time t according to the above algorithm and processing is performed.

[0065] According to the above description, for the batch processor in the periodic maintenance process, the actual problem of grouping and processing orders arriving over time is solved online. Specifically, if the machine batch capacity is sufficient, the ICD algorithm can achieve the best performance; if the machine batch capacity is limited, the FP algorithm is suitable for the maximum completion time objective, and the FPD algorithm is suitable for the maximum completion time objective, which can achieve the best performance.

[0066] Next, the performance index of the online algorithm, the competitive ratio, is described.

[0067] Let A be an online algorithm for the problem of the present patent, and I be an instance. If f A (I) represents the target value obtained by executing the algorithm A on the instance I, and f*(I) represents the optimal target value obtained by knowing all the information of the instance I. If for all instances I, f A (I)≤ρf*(I), then ρ is the competitive ratio of the algorithm. Obviously, the competitive ratio of the problem in the present patent is not less than 1. In fact, the smaller the competitive ratio, the better the performance of the algorithm. However, when the competitive ratio of the algorithm is 1, it means that the value obtained by the algorithm is equal to the optimal value, that is, the algorithm is an optimal algorithm, so the competitive ratio of the online algorithm is usually greater than 1. Therefore, the next step is to explain how good the performance of the online algorithm in the present patent can be. Hereinafter, D A and DF A represent the target value after executing the algorithm A, and D* and DF* represent the optimal target value. D j and DF j represent the target value of J j under the algorithm.

[0068] 1. Competitive ratio lower bound analysis. The best possible situation analysis of the competitive ratio of the online algorithm is called competitive ratio lower bound analysis.

[0069] First, for the makespan \(D\), max , in the case of sufficient batch capacity, when \(m = 1\), there does not exist an online algorithm with a competitive ratio less than , and when \(m>1\), there does not exist an online algorithm with a competitive ratio less than ; for the case of limited batch capacity, there does not exist an online algorithm with a competitive ratio less than \(\min\{2, 1+\alpha\}\). \(\alpha\) is the positive root of \(\alpha<**********>(should be a specific expression here, seems incomplete in the original) +2\alpha 2 -(1 + T)=1 + T\). When the transportation time is 0, the conclusion still holds.

[0070] Proof: Proved by the adversary method. The adversary gives the batch processing time \(p = 1 / 2\) and the transportation time is 0.

[0071] When the batch capacity is sufficient, and \(m + 1\) order jobs \(J_1,\cdots,J m [[ID=??]] (the number here is not clear in the original) ,J m+1 are released. And, only when the previous job is released, the next job is about to arrive. That is, if the start times of these jobs under any online algorithm \(A\) are \(S i , then the arrival time of the next job is \(r i+1 =S i +\varepsilon\). For convenience, let \(S_0 = 0\). When \(m = 1\), only need to consider Otherwise, and At this time, \(J_2\) is released. Note that \(J_2\) can only be processed after \(1 + T\) time. Therefore, \(D A \geq1 + T + p\). Since the optimal value is \(D^*=S_1 + p+\varepsilon\), so When \(m>1\), if there exists \(J j (1\leq j\leq m)\) that satisfies [[ID=??]] (the expression here is not clear in the original) Focus on the first \(j\) jobs. Assume \(J j is the first such job, then we get When \(i < j\), Therefore, we can get So all jobs can be processed in the same batch and within \([0,1]\). Then the objective value \(D\) of the optimal sequence * =r<00000**********> (the number here is not clear in the original)+p = S j-1 +p+\varepsilon\). Therefore, as \(\varepsilon\rightarrow0\), If \(J<**********> (the number here is not clear in the original)(1\leq j\leq m)\) all satisfy Then <00**********> (the number here is not clear in the original)So all jobs including \(J m+1 can form a batch and start at \(r m+1 . Then, \(D * \leq1\). However, the algorithm \(A\) cannot process \(J m+1 There seem to be some incomplete or unclear expressions in the original text which might affect the accuracy of the translation. Please check and correct the original if possible.Process within [0,1]. Therefore, D A ≥ 1 + T + p, and

[0072] When the batch capacity is limited, the adversary releases job J1 at time ε, and let p = min{α / (1 + α), 1 / 2}. Only consider S1 < min{αp, p}, otherwise, D A ≥ (1 + α)p ≥ (1 + α)D * . At this time, the adversary releases (m - 1)b + 1 jobs with transportation time 0 at time r2 = S1 + ε. Note that the optimal schedule is to process all jobs in m batches at time r2. Therefore, D * = S1 + ε + p. If S1 ≤ 1 - 2p, then D A ≥ S1 + 2p. As ε → 0, and because p = min{α / (1 + α), / 2}, we get D A / D * = 1 + p / (S1 + p) ≥ min{1 + α, 2}. If S > 1 - 2p, then D A ≥ 1 + T + p, at this time D A / D * = (1 + T + p) / min{(1 + α)p, 2p} ≥ min{1 + α, 2}.

[0073] Therefore, the conclusion holds.

[0074] Secondly, for the makespan DF max , for the case of sufficient batch capacity, there does not exist an online algorithm with a competitive ratio less than 1 + 1 / m; for the case of limited batch capacity, there does not exist an online algorithm with a competitive ratio less than 2. When the transportation time is 0, the conclusion still holds.

[0075] Proof: Proved by the adversary method. The adversary gives the batch processing time p = 1 / 2, and the job transportation time is 0. [[ID=|36]]

[0076] When the batch capacity is sufficient, release m + 1 order jobs J1,…,J m ,J m+1 , r i and S i are defined the same as in step 2.1, but define S0 = -ε. Similarly, if there exists J j satisfying S j ≥ S j-1 + (1 / m)p for 1 ≤ j ≤ m, it is easy to obtain DF A / DF<000011!6>≥ 1 + 1 / m. So, for 1 ≤ j ≤ m, only consider S j < S j-1 + (1 / m)p. Calculate Sm <S1+(m - 1)(1 / m)p, all jobs can be processed within [0, 1]. At this time, if S1 = 0, then the non-release time of the last job remains unchanged, which is r m+1 = S m + ε ≤ p. Thus, in the optimal sequence, m batches can be formed. The first batch contains J1 and J2, and the other jobs start processing in their respective batches at their release times. Then DF * = p, while DF A ≥ DF m+1 = 2p - r m+1 (1 / m)p - ε. Therefore, as ε → 0, DF A / DF * ≥ 1 + 1 / m. If S1 > 0, then the opponent releases J m+1 at time p. Therefore, J m+1 cannot be processed in the first available interval [0, 1], so DF A ≥ DF m+1 = 1 + T + p - p. However, at this time all jobs can be processed in batches within the interval [0, 1] at their release times. Thus DF * = p, DF A / DF<0000​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​

[0084] (2) Calculate LT time, note that the distance between every two LT time is 0.33p = 0.132.

[0085] Sub-interval Two LT times of [0, 0.6]: ① 0.2 - 0.132 * 1 = 0.068, ② 0.2;

[0086] Sub-interval Three LT times of [0, 1]: ① 0.6 - 0.132 * 2 = 0.336, ② 0.2 - 0.132 * 1 = 0.468, ③ 0.6;

[0087] Because the algorithm is impossible to arrange the workpiece in the third sub-interval, so it is not necessary to calculate.

[0088] (3) Group and arrange processing according to online algorithm ICD.

[0089] 1) At 0.068, workpieces J1 and J2 are released but not processed, so according to algorithm ICD, they can be processed as batch B1 at 0.068 on M2;

[0090] 2) At 0.2, only J3 can be processed, so it is alone as a batch B2 at 0.2 on the first machine M1;

[0091] 3) At 0.036, J4 can be processed as batch B3 at 0.336 on machine M3;

[0092] 4) Similarly, at 0.468 and 0.6, J5 and J6 can be processed as batches B4 and B5 at 0.468 and 0.6 on machines M2 and M1 respectively;

[0093] 5) Finally, because J7 is released at 0.7, it is already impossible to process within [0, 1], so it can only be processed within [1.2, 2.2]. The first LT time is 1.268. Therefore, J7 is processed as batch B6 at 1.268 on machine M2. The scheme is as follows Figure 4 .

[0094] 3. For the target D max and DF max under the condition of sufficient batch capacity, the competitive ratio of online algorithm ICD is 1 + 1 / m.

[0095] Proof: On the one hand, consider any available time interval, which is denoted as [l, l+1]. Observe the online algorithm ICD, it can be seen that only at the LT time can the workpiece start. In addition, it is proved that any two batches on the same machine M i+1 cannot overlap. Any two LT times of each machine can be denoted as and Here u < v. In fact, because any two LTs on the same machine are separated by at least one batch. Therefore, there are no overlapping batches.

[0096] On the other hand, any job processed in [l, l + 1] is released after time l - T - p. If [l, l + 1] is the first available interval, the conclusion is obviously true. If not, then by contradiction, assume that there is a job J j processed in [l, l + 1] and r j ≤ l - T - p. Since l - T - p is an LT, by algorithm ICD, all jobs processable at this time include J j all start at l - T - p in the same batch, which contradicts the fact that it is processed in [l, l + 1].

[0097] From the above two conclusions, the performance of online algorithm ICD can be deduced, i.e., the competitive ratio is 1 + 1 / m. Let J k be the job that reaches the target value DF k = DF ICD . Therefore, DF ICD = S k + p - r k . If it starts at the first LT in [l, l + 1], then S k < l + p / m. Since r k > l - T - p, it can only be processed in [l, l + 1] in the optimal sequence, thus DF * ≥ l + p - r k . Therefore, DF ICD - DF * ≤ p / m. And DF * ≥ p, so we get DF ICD / DF * ≤ 1 + 1 / m. If it starts at other LT in [l, l + 1], this job cannot arrive before the previous LT, otherwise it needs to be processed with other jobs in the previous LT. Since any two LTs are separated by p / m, DF ICD = S k + p - r k < p + p / m. Therefore, DF ICD / DF * ≤ 1 + 1 / m.

[0098] Similarly, when the target is D max , let J k be the job that reaches the target value, we still have D ICD - D*≤ p / m, and the conclusion holds.

[0099] When batch capacity is limited, the FP algorithm is applied.

[0100] The online algorithm FP makes full use of the idle time of all machines, with a maximum of p time lengths spent waiting for processing in each sub-interval. In Example 1, the machine capacity is set to 2, with three machines released from J1 and J2 respectively, two from J3, and one from each of the remaining jobs. The online algorithm FP is then executed; the following is the algorithm execution process.

[0101] 1) First, the start times of all sub-intervals are 0.2 and 0.6 respectively;

[0102] 2) At time 0.2, workpieces J1, J2, and J3 are released and arranged into batches according to ERD order, ultimately forming 5 batches, but only 3 batches can be processed, namely B1 = {J1, J1}, B2 = {J1, J2}, and B3 = {J2, J2}. These 3 batches are started at time 0.2.

[0103] 3) At time 0.6, there are 5 workpieces that can be machined, forming 3 batches, namely B4 = {J3, J3}, B5 = {J4, J5}, and B6 = {J6}. Start these 3 batches at time 0.6.

[0104] 4) Start the last workpiece in a separate batch at time 1.4, i.e., B6 = {J7}.

[0105] The plan is as follows Figure 5 .

[0106] For situations where machine batch capacity is limited, and the objective is D max The competition ratio for the online algorithm FP is 2.

[0107] Proof: Let J k Is it to achieve the target value D? k =D FP =C k +q k The workpieces. Note that all batches are processed within a certain sub-section of a certain machine. Let's assume J... k In subinterval Processing. Before this subinterval, find the last subinterval that satisfies the following conditions. (a) There exists a machine in the sub-interval (a) Idle; (b) There are non-full batches in the sub-intervals. Processing. Additionally, define a set W of workpieces, which consists of all workpieces at time [time]. arrive Internally machined workpieces include J k Composition. Because J kis the last processed job, so according to the ERD rule in the algorithm, r k is the last release time in the W job set. In fact, according to the algorithm FP, the release time of all jobs in the W set is greater than For the optimal sequence, define J e ∈ W is the last job in the set W to start in the optimal sequence.

[0108] First, if the subinterval does not exist, then all the subintervals are full of jobs, so we can get that in the optimal sequence, there are at most C k - C e* ≤ xp < p, where C e* is J e the completion time on the machine in the optimal sequence. Next, consider the case that exists, and discuss it in two cases.

[0109] Case 1. h = l.

[0110] Since there are m(u k - v - 1)b + 1 jobs in W, so For the optimal sequence, we have Therefore,

[0111] Case 2. h < l. Assume that there are g available time intervals between h and l, so there are at least m(y - v + yg + u k - 1)b + 1 jobs in W.

[0112] If yp = 1, then C k = l + u k p. Since the release time of all jobs is greater than so we can get that C e* ≥ l + u k p. Therefore, C k - C e* ≤ 0. If yp < 1, then At least m(u k - 1)b + 1 jobs start after l, so C e* ≥ l + u k p. Thus,

[0113] In summary, D FP - D * ≤ C k - C e* + q k - q e ≤ p + q k ≤ D * . Therefore, DFP / D * ≤2. The conclusion is valid.

[0114] 4. Online algorithm FPD analysis.

[0115] Below, we will set the machine capacity in Example 1 to 2, release three J1 and J2, release two J3, and release one of each of the remaining workpieces, and increase their transportation time to generate Example 2.

[0116]

Example 2

[0117] The data for 7 workpieces with a capacity of 2 is shown in Table 3 after adding the transportation time, where p = 0.4 and T = 0.1.

[0118] Table 3 Workpiece Release Time

[0119]

[0120]

[0121] Based on the execution steps of the FPD algorithm, the structure of the final generated batch remains the same. Figure 5 As shown. However, some batches contain different workpieces. Workpiece priorities are calculated: J1 is 2, J2 is 0.97, J3 is 3.98, J4 is 4.75, J5 is 1.55, J6 is 3.4, and J7 is 0.3. Their indices are then sorted: 4→3→6→1→5→2→7. Thus, seven batches are formed: B1={J3,J3}, B2={J1,J1}, B3={J1,J2}, B4={J4,J6}, B5={J5,J2}, B6={J2}, B6={J7}.

[0122] For situations where machine batch capacity is limited, the target is DF max The competition ratio of the online algorithm FPD is 2.

[0123] Proof: Let J k Is it to achieve the target value DF? k =DF FPD =C k +q k -r k The workpieces. All batches are still processed within a specific sub-section of a certain machine. Again, assume J... k In subinterval Processing. Before this subinterval, find the last subinterval that satisfies the following conditions. (a) There exists a machine in the sub-interval (a) Idle; (b) There are non-full batches in the sub-intervals. Processing; (c) There exists a workpiece in this sub-interval with a priority lower than J.k In addition, define the set of workpieces W, which consists of all workpieces processed in the time interval k In fact, according to the algorithm FPD, the release time of all workpieces in the set W is greater than For the optimal sequence, define J e ∈ W is the last workpiece started in the set W under the optimal sequence.

[0124] First, if the sub-interval does not exist, then all sub-intervals are filled with workpieces, and it is proved in step 3.2 that C k -C e* ≤ xp < p. In addition, because DF * = C e* + q e - r e ≥ C e* + q k - r k , DF FPD - DF * ≤ p + q k - r k ≤ DF * . Therefore, D FPD / D * ≤ 2. The conclusion is true.

[0125] Therefore, if the machine batch capacity is sufficient, the ICD algorithm can achieve the best performance; if the machine batch capacity is limited, the FP algorithm is suitable for the maximum completion time target, and the FPD algorithm is suitable for the maximum completion time target, which can all achieve the best performance.

[0126] Although the embodiments of the present application have been shown and described above, it is understood that the above-described embodiments are exemplary and are not to be construed as limiting the present application, and those of ordinary skill in the art can make changes, modifications, replacements and variations to the above-described embodiments without departing from the principles and purposes of the present application within the scope of the present application.​​

Claims

1. An online scheduling method considering periodic maintenance of batch processors, characterized in that: The method comprises the following steps: Step 1: mathematical modeling of the on-line scheduling problem of batch machines under periodic maintenance: The established model is: Order job J j Arrival over time, its arrival, the job information is unknown; The workpieces need to be processed on m batch machines, the set of machines is denoted as {M1, M2, …, M m} The batch machine can process b workpieces at the same time, and the processing time of each batch workpiece is fixed as 0 < p < 1; All machines need periodic regular maintenance, and the maintenance time length is T; during the maintenance process, the machine cannot process workpieces; Let the machine available time between two maintenances be set as a unit time 1, and the machine available time interval is denoted as [l k , k +1], where l k+1 =0, and l k =l + +1+T, k∈N Each workpiece is completed at a completion time C j The customer needs the workpiece immediately after completion, and the delivery time is noted as q j , and the vehicle is sufficient; only when the workpiece is delivered to the customer does the workpiece flow end Optimization scheduling goal is to minimize D max = max j {C j + q j}, DF max = max j {C j + q j - r j} ; wherein the arrival time of workpiece J j r j , the completion time C j , the transportation time is recorded as q j ; D max , that is, the maximum transportation completion time, DF max , that is, the maximum process time; Step 2: obtaining the current time t, the number of machines m, the batch capacity b, the batch processing time p, and the information of the workpieces that have arrived but not processed at the current time, including the arrival time and the transportation time; Step 3: processing each available time interval as follows: Let [l, l+1] be any available time interval; Set x, y, satisfying xp+yp=1, wherein 0≤x<1, y≥1 is a positive integer; Setting sub-interval Is: Step 3.1: when the batch capacity is sufficient, the on-line algorithm Infinite Capacity Delay (ICD) is used for solving; The online algorithm ICD is: setting LT moment in each interval: Except the first interval, each sub-interval with length p has m LT moments; if the current moment t is a certain LT moment And if the machine is free, then all workpieces at this moment form a batch on machine M i+1 Upper processing; otherwise, wait until the next LT moment; Step 3.2: When the batch capacity is limited and the optimization goal is D max solved by applying the Full Processing (FP) algorithm. The FP algorithm is as follows: if the current time t is the start time of a subinterval of length p... If there are idle machines at that moment, all workpieces should be grouped into full batches according to the rule of minimum release time priority; assuming there are 'a' batches, and the first 'a-1' batches are all full batches; then the first min{m,a} batches are placed at time [time value missing]. Processing on any idle machine; Step 3.3: In case of limited batch capacity and optimization goal is DF max , apply Full Processing under Delivery (FPD) algorithm for solution. The FPD algorithm is: if the current time t is the starting time of a sub-interval with length p , and the machine is idle at this time, then all workpieces are grouped into full batches as much as possible according to the D-ERD rule; assuming that a batches are formed, and the first a-1 batches are full batches, then the first min{m, a} batches are placed at time , and the workpieces are processed on any idle machine; the D-ERD rule is that workpieces are sorted in non-decreasing order of q j -r j , and if there are multiple workpieces with the same q j -r j , then the workpieces are sorted again according to the ERD rule. Step 4: at time t, using the algorithm described in step 3 to judge whether batch processing is performed; if the time t meets the processing condition, the scheduling scheme is output at the current time t according to the above algorithm and processing is performed.

2. The online scheduling method considering periodic maintenance of batch processors according to claim 1, wherein: In Step 3, for the maximum makespan D max For the case of unlimited batch size, when m = 1, there is no online algorithm with a competitive ratio less than For the case of unlimited batch size, when m = 1, there is no online algorithm with a competitive ratio less than For the case of unlimited batch size, when m = 1, there is no online algorithm with a competitive ratio less than 3 For the case of unlimited batch size, when m = 1, there is no online algorithm with a competitive ratio less than 2 For the case of unlimited batch size, when m = 1, there is no online algorithm with a competitive ratio less than max For the case of unlimited batch size, when m = 1, there is no online algorithm with a competitive ratio less than 3. The online scheduling method considering periodic maintenance of batch process according to claim 2, characterized in that: In Step 3, for the case where the batch capacity is sufficient for the target D max and DF max The competitive ratio of the online algorithm ICD is 1+1 / m.

4. The online scheduling method considering periodic maintenance of batch process according to claim 2, characterized in that: In Step 3, for the case of limited machine batch capacity, and the target is D max The competitive ratio of the online algorithm FP is 2.

5. The online scheduling method considering periodic maintenance of batch process according to claim 2, characterized in that: In Step 3, for the case of limited machine batch capacity, the goal is DF max The competitive ratio of the online algorithm FPD is 2.

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