CN107967558B - Input-output decision method considering cost of parallel batch processors - Google Patents

Input-output decision method considering cost of parallel batch processors Download PDF

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CN107967558B
CN107967558B CN201711154748.7A CN201711154748A CN107967558B CN 107967558 B CN107967558 B CN 107967558B CN 201711154748 A CN201711154748 A CN 201711154748A CN 107967558 B CN107967558 B CN 107967558B
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王军强
胥军
王艳
范国强
闫飞一
崔鹏浩
夏添
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Abstract

The invention provides an input-output decision method considering the cost of a parallel batch processor, which takes the parallel batch processor as a research object, considers the practical problem of production cost increase caused by the capacity change of the batch processor, designs an effective heuristic algorithm by establishing a function of cost increment and the capacity and the quantity of the batch processor, balances the cost and the benefit, performs input-output analysis and obtains a batch processor capacity expansion decision scheme. The invention aims to provide a theoretical basis for decisions such as transformation of old line equipment of an enterprise, planning of new line capacity and the like and also provides a referential research mode for upgrading and upgrading of other types of production lines, capacity improvement of the production lines and the like.

Description

Input-output decision method considering cost of parallel batch processors
Technical Field
The invention relates to the technical field of batch processing scheduling in a production workshop, in particular to an input-output decision problem considering the cost of a parallel batch processor.
Background
Batch processors (Batch Processing machines) expand the single-piece Processing mode to the Batch Processing mode, and break through the limitation that one Machine can only input or output one workpiece at a time. The batch processor has the advantages of large early investment, long construction period, high operation and maintenance cost, large energy consumption and small quantity of construction, is generally a bottleneck machine of a production workshop, is also a key unit for pulling and pushing material flow of a manufacturing system upwards and downwards, and is a place where the production line has high input, high output and high added value. Production Scheduling (Scheduling) is a neural center which ensures ordered, stable, balanced and efficient operation of a production process, and is a key link which restricts the performance of a production system. The traditional production scheduling research focuses on seeking an optimization scheme, but the production workshop has complex processes and multiple constraint conditions, so that the optimization space is small and the optimization difficulty is high.
The parallel batch scheduling problem can be described as: given a workpiece set J containing n workpieces { J ═ J1,J2,…,JnSet of machines with M parallel batch processors M ═ M1,M2,…,Mm}, workpiece JjHas a processing time of pjThe workpiece lot set B ═ B1,B2,…,Bk… }, batch k processing time p (B)k) Arranged in a machine MiK (k) th batch processing time p (B)ik). Parallel batch processor M in the scheduling problem of batch processors in a parallel machine environmentiAt most b workpieces can be processed at the same time, the workpieces arrive at the time 0, the batch processor starts to process at the time 0, no preparation time exists between each batch of workpieces, and the processing process cannot be interrupted.
The parallel batch processing scheduling problem is described by a mathematical language to be Pm | p-batch, b is more than n | Cmax，CmaxThe maximum completion time is an optimization target; the mathematical model is as follows
obj.minCmax (1)
Figure GDA0001559085640000011
Figure GDA0001559085640000012
p(Bik)≥xjikpj j＝1,2,…,n，i＝1,2,…,m， (4)
C(Bik)≥C(Bi(k-1))+p(Bik)i＝1,2,…,m， (5)
Figure GDA0001559085640000021
xjik∈{0,1}i＝1,2,…,m，j＝1,2,…,n (7)
Wherein x isjikIs a variable of 0 to 1, when the workpiece JjAt machine MiIs processed and arranged at BkIn a batch, thenxjik1, otherwise xjik＝0；C(Bik) Indicating machine MiThe time for completion of the upper kth workpiece.
Equation (1) is the objective function of the model, i.e. minimizing the maximum completion time; formula (2) indicates that each workpiece can only be assembled into one batch and processed on one machine; the formula (3) shows that the number of workpieces in each batch of the batch processor does not exceed the capacity b of the batch processor; the formula (4) represents that the processing time of each batch of workpieces is not less than that of any workpiece in the batch; the formula (5) shows that the finishing time of each batch of workpieces is not less than the sum of the finishing time of the previous batch of workpieces and the processing time of the batch of workpieces; the expression (6) indicates that the maximum completion time is not less than the maximum completion time of each batch processor; formula (7) represents xjikIs a variable from 0 to 1.
Disclosure of Invention
Technical problem to be solved
Batch processor expansion is intended to increase manufacturing plant capacity to maximize profits. Considering the cost of a batch processor, analyzing input and output, balancing the cost of the batch processor and the maximum completion time of a workpiece, obtaining a decision scheme of batch capacity after the batch processor expands the capacity, providing theoretical support for upgrading and transforming workshop equipment, and solving the technical problem of the patent.
The invention takes a parallel batch processor as a research object, considers the practical problem of production cost increase caused by the capacity change of the batch processor, designs an effective heuristic algorithm by establishing a function of cost increment and the capacity and the quantity of the batch processor, balances cost and benefit, performs input-output analysis and obtains a batch processor capacity expansion decision scheme. The invention aims to provide a theoretical basis for decisions such as transformation of old line equipment of an enterprise, planning of new line capacity and the like and also provides a referential research mode for upgrading and upgrading of other types of production lines, capacity improvement of the production lines and the like.
The technical scheme of the invention is as follows:
the input-output decision method considering the cost of the parallel batch processor is characterized by comprising the following steps: the method comprises the following steps:
step 1: establishing an input-output decision problem mathematical model of the capacity cost of the parallel batch processor:
the parallel batch processing machines are used for processing n workpieces, and the parallel batch processing machines can process b workpieces at most at the same time; the optimization objective is to minimize the maximum machining time CmaxSum of batch processor capacity cost:
Figure GDA0001559085640000022
beta is a batch processing capacity cost coefficient;
the batch scheduling problem is represented as: pm | p-batch, b < n | Cmax+βmb
The mathematical model is as follows:
obj.min(Cmax+βmb)
Figure GDA0001559085640000031
Figure GDA0001559085640000032
p(Bik)≥xjikpj j＝1,2,…,n，i＝1,2,…,m
C(Bik)≥C(Bi(k-1))+p(Bik)i＝1,2,…,m
Figure GDA0001559085640000033
xjik∈{0,1}i＝1,2,…,m，j＝1,2,…,n
wherein x isjikIs a variable of 0 to 1, when the workpiece JjAt machine MiIs processed and arranged at BkIn a batch, then xjik1, otherwise xjik＝0；
Step 2: loosening the workpiece to be interruptible to obtain a loosening problem of the original problem, and analyzing the optimal solution property of the loosening problem:
the batch scheduling problem Pm | p-batch established in step 1, b < n | CmaxAdding a constraint condition that the workpiece can be interrupted in the + beta mb to obtain a batch processing scheduling questionThe relaxation problem is Pm | p-batch, b < n, pmtn | Cmax+βmb；
For the relaxation problem Pm | p-batch, b < n, pmtn | CmaxCarrying out optimal solution on the beta mb to obtain the optimal solution
Figure GDA0001559085640000034
Wherein b is*Representing an objective function
Figure GDA0001559085640000035
Minimum batch processing capacity; s*(b*) Denotes batch processing capacity b*A time optimal scheduling scheme; cmax(S*(b*) Represents the maximum completion time under the optimal scheduling scheme;
a cost function of
Figure GDA0001559085640000036
Where the subscript p indicates that the workpiece can be interrupted,
Figure GDA0001559085640000037
when the batch processing capacity is b, the optimal scheduling scheme under the condition that the workpieces can be interrupted is represented, and beta represents a batch processing capacity cost coefficient; the cost function is expressed as:
Figure GDA0001559085640000038
wherein f is1(b)＝pmax+βmb，f2(b)＝Pmb+βmb；
And step 3: designing a heuristic algorithm 1 according to the optimal solution property of the relaxation problem, and solving the optimal batch processing capacity of the relaxation problem:
order to
Figure GDA0001559085640000041
Representing the optimal batch processing capacity when a workpiece can be interrupted, and representing the optimal cost function of the relaxation problem as
Figure GDA0001559085640000042
f1(b) Is a monotonically increasing function of batch processing capacity b, resulting in minf1(b)＝f1(b1) (ii) a Cost function f2(b) Is a convex function when
Figure GDA0001559085640000043
The time objective function obtains global optimum, when b is less than or equal to mu, f2(b) Is a monotone decreasing function, when b is more than or equal to mu, f2(b) Is a monotonically increasing function, obtaining
Figure GDA0001559085640000044
Thereby obtaining
Figure GDA0001559085640000045
Aiming at the relaxation problem Pm | p-batch, b is less than n,
Figure GDA00015590856400000418
solving the batch processing capacity b for minimizing the cost function by the following steps1(ii) a Wherein the batch processing capacity b1To enable the maximum processing time of a workpiece to be not less than the minimum batch processing capacity at the average load of the machines for a given number of machines, p is satisfiedmax≥Pmb1Minimum batch processing capacity of
Figure GDA00015590856400000419
Wherein
Figure GDA0001559085640000046
Step 3.1: by the equation
Figure GDA00015590856400000420
Calculation of b1And make an order
Figure GDA00015590856400000421
Step 3.2: by the equation
Figure GDA0001559085640000047
Calculating mu;
step 3.2.1: if it is
Figure GDA0001559085640000048
Go to step 3.3, otherwise go to step 3.2;
step 3.2.2: let b2＝b11, if f2(b2)≤f1(b1) Then, then
Figure GDA0001559085640000049
Otherwise
Figure GDA00015590856400000410
Go to step 3.4;
step 3.3: computing
Figure GDA00015590856400000411
And
Figure GDA00015590856400000412
if f2(b4)≤f2(b3)≤f1(b1) Then, then
Figure GDA00015590856400000413
If f2(b3)≤f2(b4)≤f1(b1) Then, then
Figure GDA00015590856400000414
Otherwise
Figure GDA00015590856400000415
Step 3.4: output of
Figure GDA00015590856400000416
And
Figure GDA00015590856400000417
terminating;
and 4, step 4: designing a heuristic algorithm RPH, and solving a total cost function of the relaxation problem under the optimal batch processing capacity:
analyzing a scheduling problem considering batch processing capacity cost under the condition that workpieces cannot be interrupted
Pm|p-batch,b＜n|Cmax+βmb
The objective function after considering the cost of the batch processing capacity is
Figure GDA0001559085640000051
Where the subscript np indicates that the workpiece is not interruptible,
Figure GDA0001559085640000052
representing the optimal scheduling scheme under the condition that the workpieces cannot be interrupted when the batch processing capacity is b;
get
Figure GDA0001559085640000053
Represents the optimal batch processing capacity under the condition that workpieces cannot be interrupted, and has all batch processing capacities b more than or equal to 1
Figure GDA0001559085640000054
If true; under the condition of the same batch processing capacity, the upper bound of the ratio of the optimal target value when the workpiece can not be interrupted to the optimal target value when the workpiece can be interrupted is 2-1 mb;
the following heuristic algorithm is adopted to make a scheduling scheme:
step 4.1: obtaining the optimal batch processing capacity for the relaxation problem according to step 3
Figure GDA0001559085640000055
Order to
Figure GDA0001559085640000056
Step 4.2: according to b, obtained in step 4.1The following heuristic algorithm H1 outputs a scheduling scheme Cmax(Snp(b) ) and cost function
Figure GDA0001559085640000057
Step 4.2.1: workpiece collection
Figure GDA0001559085640000058
All workpieces are batched according to the FBLPT rule to obtain k batches,
Figure GDA0001559085640000059
step 4.2.2: each batch of workpieces is scheduled on the machine that minimizes its completion time according to LPT rules until all batches are scheduled.
Advantageous effects
In order to verify the rationality of the proposed RPH algorithm, an experiment for taking cost coefficients of different batch processing capacities under the condition of fixed machine number is provided, and specific parameters are shown in Table 1. By simulation experiments, the error magnitude of the heuristic algorithm RPH and the lower bound of the optimal solution is compared, and the following conclusion can be obtained from the graph 1:
1) the method provides a scheduling problem of batch processor capacity expansion, considers the cost required by the batch processor capacity expansion, designs an algorithm, and solves the optimal batch processing capacity, thereby further expanding the batch processing scheduling problem.
2) The maximum deviation of the heuristic algorithm RPH and the optimal solution is not more than 5%, and the average deviation is in a descending trend along with the increase of the example scale;
3) comparing subgraphs, wherein when the calculation examples are the same in scale, the larger the cost coefficient of the batch processing capacity is, the smaller the average deviation is;
4) and (4) comparing in a subgraph, wherein the average deviation is gradually reduced along with the increase of the scale of the calculation example when the batch processing capacity cost coefficients are the same.
According to the theoretical result, b < n | C for the problem Pm | p-batchmax+ β mb, the worst performance ratio of the heuristic RPH does not exceed 2 at most. According to the experimental result, the heuristic algorithm RPH is the mostThe large deviation does not exceed 5%, and gradually decreases as the scale of the example increases.
Additional aspects and advantages of the invention will be set forth in part in the description which follows and, in part, will be obvious from the description, or may be learned by practice of the invention.
Drawings
The above and/or additional aspects and advantages of the present invention will become apparent and readily appreciated from the following description of the embodiments, taken in conjunction with the accompanying drawings of which:
FIG. 1 results of an algorithm RPH experiment;
FIG. 2 Algorithm H1 Gantt chart;
FIG. 3 is a functional diagram of an example.
Detailed Description
The following detailed description of embodiments of the invention is intended to be illustrative, and not to be construed as limiting the invention.
This patent presents an input-output decision problem that takes into account the cost of parallel batch processors and explains their rationality through simulation experiments. The input-output decision method considering the cost of the parallel batch processor is mainly characterized by comprising the following steps: providing a mathematical model of the relevant problem; designing a cost function of a batch processor, and giving an optimal solution of the relaxation problem; designing a heuristic algorithm RPH; analyzing the performance of a heuristic algorithm RPH; giving an optimal solution lower bound algorithm; and designing an experimental simulation of an example, and verifying the validity of a heuristic algorithm and the rationality of the method.
Step 1: and establishing an input-output decision problem mathematical model of the capacity cost of the parallel batch processor. m parallel batch processors with the same processing speed process n workpieces, and the optimization aim is to minimize the sum of the maximum processing time and the capacity cost of the batch processors.
The optimization objective is the sum of the maximum processing time and the batch processing capacity cost, i.e.
Figure GDA0001559085640000061
Where β is the batch capacity cost factor. The batch scheduling problem studied is represented as: pmp | p-batch,b＜n|Cmax+ β mb, mathematical model as follows:
obj.min(Cmax+βmb) (8)
Figure GDA0001559085640000071
Figure GDA0001559085640000072
p(Bik)≥xjikpj j＝1,2,…,n，i＝1,2,…,m (11)
C(Bik)≥C(Bi(k-1))+p(Bik)i＝1,2,…,m， (12)
Figure GDA0001559085640000073
xjik∈{0,1}i＝1,2,…,m，j＝1,2,…,n (14)
wherein x isjikIs a variable of 0 to 1, when the workpiece JjAt machine MiIs processed and arranged at BkIn a batch, then xjik1, otherwise xjik＝0。
Equation (8) is the objective function of the model, i.e., the minimization of the total cost function; formula (9) indicates that each workpiece can only be grouped into one batch and processed on one machine; the formula (10) shows that the number of workpieces in each batch of the batch processor does not exceed the capacity b of the batch processor; formula (11) represents that the processing time of each batch of workpieces is not less than the processing time of any one workpiece in the batch; the formula (12) shows that the finishing time of each batch of workpieces is not less than the sum of the finishing time of the previous batch of workpieces and the processing time of the batch of workpieces; equation (13) represents that the maximum completion time is not less than the maximum completion time of each batch processor; formula (14) represents xjikIs a variable from 0 to 1.
Step 2: and (4) analyzing an optimal solution of the relaxation problem. And loosening the workpiece to be interruptible to obtain the loosening problem of the original problem, and analyzing the optimal solution property of the loosening problem.
Batch tone created in step 1Degree problem Pm | p-batch, b < n | CmaxAdding constraint condition that workpiece can be interrupted in + beta mb to obtain relaxation problem of batch scheduling problem Pm | p-batch, b < n, pmtn | Cmax+ β mb; for the problem of relaxation
Pm|p-batch,b＜n,pmtn|Cmax+βmb
Carrying out optimal solution to obtain the optimal solution
Figure GDA0001559085640000074
Wherein b is*Representing an objective function
Figure GDA0001559085640000075
Minimum batch processing capacity; s*(b*) Denotes batch processing capacity b*A time optimal scheduling scheme; cmax(S*(b*) Represents the maximum completion time under the optimal scheduling scheme.
Cost function
Figure GDA0001559085640000076
Where the subscript p indicates that the workpiece can be interrupted,
Figure GDA0001559085640000077
and when the batch processing capacity is b, the optimal scheduling scheme under the condition that the workpieces can be interrupted is shown, and beta represents a batch processing capacity cost coefficient. The cost function can be expressed as follows
Figure GDA0001559085640000081
Wherein f is1(b)＝pmax+βmb，f2(b)＝Pmb+βmb。
And step 3: and solving the optimal batch processing capacity of the relaxation problem. And designing a heuristic algorithm 1 according to the optimal solution property of the relaxation problem, and solving the optimal batch processing capacity of the relaxation problem.
Order to
Figure GDA0001559085640000082
Indicating workpiece may be inThe optimal cost function for the time-out optimal batch processing capacity, relaxation problem can be expressed as
Figure GDA0001559085640000083
Due to f1(b) Is a monotonically increasing function with respect to batch processing capacity b, so minf1(b)＝f1(b1)。
Cost function f2(b) Is a convex function, the variation trend is similar to that of the economic order batch (EOQ) model
Figure GDA0001559085640000084
The time objective function obtains global optimum, when b is less than or equal to mu, f2(b) Is a monotone decreasing function, when b is more than or equal to mu, f2(b) Is a monotonically increasing function. Then there is
Figure GDA0001559085640000085
Further, it can be obtained
Figure GDA0001559085640000086
Aiming at the relaxation problem Pm | p-batch, b < n and pmtn | Cmax+ β mb, defining the batch capacity b at a given number of machines1To enable the maximum processing time of the workpiece to be not less than the minimum batch capacity at the average load of the machine, i.e. to satisfy pmax≥Pmb1Minimum batch processing capacity of
Figure GDA00015590856400000813
Wherein
Figure GDA0001559085640000087
Aiming at the relaxation problem Pm | p-batch, b is less than n,
Figure GDA00015590856400000814
algorithm 1 was designed to solveThe cost function minimum batch processing capacity, algorithm 1 is specified below.
Inputting an algorithm: workpiece collection
Figure GDA0001559085640000089
Set of batch processors
Figure GDA00015590856400000810
Batch capacity cost factor β.
And (3) outputting an algorithm: optimal batch processing capacity for relaxation issues
Figure GDA00015590856400000811
Step 3.1: by the equation
Figure GDA00015590856400000815
Calculation of b1And make an order
Figure GDA00015590856400000812
Step 3.2: by the equation
Figure GDA00015590856400000920
Calculating mu;
step 3.2.1: if it is
Figure GDA0001559085640000092
Go to step 3.3, otherwise go to step 3.2;
step 3.2.2: let b2＝b 11, if f2(b2)≤f1(b1) Then, then
Figure GDA0001559085640000093
Otherwise
Figure GDA0001559085640000094
Go to step 3.4;
step 3.3: computing
Figure GDA0001559085640000095
And
Figure GDA0001559085640000096
if f2(b4)≤f2(b3)≤f1(b1) Then, then
Figure GDA0001559085640000097
If f2(b3)≤f2(b4)≤f1(b1) Then, then
Figure GDA0001559085640000098
Otherwise
Figure GDA0001559085640000099
Step 3.4: output of
Figure GDA00015590856400000910
And
Figure GDA00015590856400000911
and (6) terminating.
And 4, step 4: and designing a heuristic algorithm RPH, and solving a total cost function of the relaxation problem under the optimal batch processing capacity.
In the actual production process, the workpiece is not allowed to be interrupted in the batch processing process, and the scheduling problem of batch processing capacity cost is analyzed and considered under the condition that the workpiece is not interrupted, namely Pmp-batch, b < n Cmax+βmb。
The objective function after considering the cost of the batch processing capacity is
Figure GDA00015590856400000912
Where the subscript np indicates that the workpiece is not interruptible,
Figure GDA00015590856400000913
and the optimal scheduling scheme is shown under the condition that the workpieces cannot be interrupted when the batch processing capacity is b.
Order to
Figure GDA00015590856400000914
Indicating the optimal batch processing capacity in the case of uninterrupted workpieces, i.e. for all batch processing capacities b ≧ 1
Figure GDA00015590856400000915
This is true.
The algorithm designed in this subsection needs to analyze the ratio of the optimal target value when the workpiece is not interruptible to the optimal target value when the workpiece is interruptible under the condition of the same batch processing capacity, so the influence of the workpiece interruptible on the target function under the environment of the parallel batch processor is analyzed first.
Step 4.1: the effect of the workpiece interrupt on the objective function.
Aiming at the problem Pm | | CmaxThe influence of workpiece interruption on the objective function is measured by the following indexes
Figure GDA00015590856400000916
Braun et al (2003) and Lee et al (2005) gave the following conclusions:
Figure GDA00015590856400000917
braun et al (2003) have demonstrated an optimal solution in the case of uninterruptable workpieces
Figure GDA00015590856400000918
Results obtained with LPT Algorithm
Figure GDA00015590856400000919
Instead, the above equation is also true. The following analyzes the impact of workpiece interruption on the objective function in the parallel batch scheduling problem in the parallel machine environment.
It can be demonstrated that the problem Pm | p-batch, b < n | CmaxThe optimal solution and problem Pm | p-batch, b < n, pmtn | CmaxThe upper bound of the optimal solution ratio is 2-1 mb.
Step 4.2: and designing a heuristic algorithm RPH.
The heuristic algorithm RPH design idea is as follows: the optimal batch processing capacity of the relaxation problem is solved through the algorithm 1, and a scheduling scheme is formulated by adopting a heuristic algorithm H1 based on the optimal batch processing capacity solved by the algorithm 1. The heuristic algorithm RPH is as follows.
Inputting an algorithm: workpiece collection
Figure GDA0001559085640000101
Set of batch processors
Figure GDA0001559085640000102
Batch capacity cost factor β.
And (3) outputting an algorithm: batch processing capacity b, scheduling scheme Cmax(Snp(b) Cost function), cost function
Figure GDA0001559085640000103
Step 4.2.1: running Algorithm 1 to find the optimal batch Capacity for relaxation issues
Figure GDA0001559085640000104
Order to
Figure GDA0001559085640000105
Step 4.2.2: a heuristic algorithm H1(FBLPT) is introduced.
The scheduling problem of the parallel batch processor in the parallel machine environment relates to two subproblems, namely, how to group workpieces and how to sequence the grouped batches. Regardless of the volume cost of the batch processor, for the problem Pmp-batch, b < n CmaxA heuristic algorithm H1(FBLPT) is introduced.
Step 4.2.2.1: workpiece collection
Figure GDA0001559085640000106
All the workpieces are batched according to the rule of FBLPT (full Batch lost Processing time), so that k batches can be obtained, namely
Figure GDA0001559085640000107
Step 4.2.2.2: each batch of workpieces is scheduled on the machine that minimizes its completion time according to the lpt (long Processing time) rule until all batches are scheduled.
Step 4.2.3: operating a heuristic algorithm H1 according to the b obtained in the step 4.2.1, and outputting Cmax(Snp(b) ) and
Figure GDA0001559085640000108
[ example 1 ]
Two parallel batch processors with a batch capacity of b 2, 6 workpiece processing times are shown in table 1.
TABLE 1 workpiece machining time
Figure GDA0001559085640000109
According to heuristic H1, the batch scheme is B1＝{J1,J2}，B2＝{J3,J4}，B3＝{J5,J6The scheduling scheme is machine M1Upper processing batch B2、B3Machine M2Upper processing batch B1. The maximum completion time of the workpiece is thus Cmax＝max{C1,C2}＝max{p(B1),p(B2)+p(B3)}＝max{11,13}＝13。
The gantt chart of heuristic H1 is shown in FIG. 2.
And 5: and (5) carrying out RPH performance analysis by a heuristic algorithm.
The performance of the heuristic RPH depends greatly on the quality
Figure GDA00015590856400001114
The difference size, and therefore the performance analysis of the heuristic RPH is divided into the following two parts.
Step 5.1: heuristic algorithm RPH performance analysis case 1.
For the solution in Algorithm 1
Figure GDA0001559085640000111
Heuristic algorithm RPH worst performance ratio is
Figure GDA0001559085640000112
And (3) proving that: the optimal solution when the workpiece can not be interrupted is not less than that when the workpiece can be interrupted, namely the following inequality relation is satisfied
Figure GDA0001559085640000113
The optimal batch processing capacity value is only two conditions when the workpiece can be interrupted, and the worst performance ratio of the heuristic algorithm RPH under the two conditions is respectively analyzed.
Case 1 if
Figure GDA0001559085640000114
When there is
Figure GDA0001559085640000115
Case 2 if
Figure GDA0001559085640000116
When the temperature of the water is higher than the set temperature,
Figure GDA0001559085640000117
then
Figure GDA0001559085640000118
So that there are
Figure GDA0001559085640000119
Thus, it is possible to provide
Figure GDA00015590856400001110
When in use
Figure GDA00015590856400001111
In time, the batch processing capacity cost beta mb contributes more to the cost function, so that the worst performance ratio with better performance of the heuristic algorithm RPH is analyzed in the next step.
Step 5.2: heuristic algorithm RPH performance analysis case 2.
For the solution in Algorithm 1
Figure GDA00015590856400001112
Heuristic algorithm RPH worst performance ratio is
Figure GDA00015590856400001113
And (3) proving that: discussed in two cases below.
Case 1 if
Figure GDA0001559085640000121
When the temperature of the water is higher than the set temperature,
Figure GDA0001559085640000122
so that there are
Figure GDA0001559085640000123
Due to the fact that
Figure GDA0001559085640000124
Can obtain the product
Figure GDA0001559085640000125
By substituting formula (26) for formula (25), a compound having the formula
Figure GDA0001559085640000126
Case 2 if
Figure GDA0001559085640000127
When the temperature of the water is higher than the set temperature,
Figure GDA0001559085640000128
so that there are
Figure GDA0001559085640000129
Due to the fact that
Figure GDA00015590856400001210
Can obtain the product
Figure GDA00015590856400001211
From the formulae (28) and (29), it is possible to obtain
Figure GDA00015590856400001212
Thus, it is possible to provide
Figure GDA0001559085640000131
Step 6: and (4) an optimal solution lower bound algorithm.
Problem Pm | p-batch, b < n | Cmax+ mb is NP-hard, and there is no polynomial time optimal algorithm, so this section works out the lower bound of the optimal solution by designing the lower bound algorithm, and then performs the simulation experiment on it.
Aiming at the problem Pm | p-batch, b < n | Cmax+ beta mb, slackening the workpiece to be interruptible, and designing an optimal solution lower bound algorithm of the original problem by using the optimal solution of the slackening problem, wherein the specific steps are as follows.
Step 6.1: batch capacity b is 1, if
Figure GDA0001559085640000132
Go to step 6.2, otherwise go to step 6.6;
step 6.2: will be assembled
Figure GDA0001559085640000133
The medium workpieces are batched according to the FBLPT rule to obtain k batches of workpieces, and the set of all batches is
Figure GDA0001559085640000134
Step 6.3: if k is less than or equal to 2m, the sets are collected
Figure GDA0001559085640000135
Each batch is regarded as a 'workpiece', and k batches of workpieces are arranged on m machines in turn according to LPT rules, and the obtained solution is the optimal solution. Otherwise, go to step 6.4;
step 6.4: if k is greater than or equal to 2m, the set is collected
Figure GDA0001559085640000139
All batches of workpieces in (1) are distributed to each machine evenly;
step 6.5: substituting the maximum completion time and the batch processing capacity into a cost function, calculating a cost function value, and returning to the step 6.1 when b is equal to b + 1;
step 6.6: and outputting the batch processing capacity and the minimum cost function value.
Aiming at the problem Pm | p-batch, b < n | Cmax+ β mb, completion time LB for the lower bound algorithm may be expressed as follows
Figure GDA0001559085640000136
Wherein
Figure GDA0001559085640000137
Indicating the number of batches grouped.
And 7: and (5) analyzing the rationality.
In order to measure the performance of the heuristic RPH, for any one example I, the solution obtained by the heuristic RPH is represented by RPH (I), the optimal solution is represented by OPT (I), and the lower bound of the optimal solution is represented by LB (I). Defining the heuristic algorithm RPH with a relative deviation dev (RPH) of
Figure GDA0001559085640000138
For any one of examples I, LB (I). ltoreq.OPT (I) holds, so that the solution obtained by the heuristic RPH does not deviate from the optimal solution by more than dev (RPH), i.e.
Figure GDA0001559085640000141
Each figure includes four subgraphs (a), (b), (c) and (d), and shows the variation of the deviation when the batch beta is 0.5,1,1.5 and 2. The abscissa in each sub-graph represents the number of workpieces from 100 to 1000 and the ordinate represents the deviation dev (RPH) of the heuristic from the optimal solution lower bound. Each graph contains the magnitude of the deviation for different batch volume increments. Each data point is the average of 100 arithmetic deviations. The smaller the deviation, the better the heuristic performance.
Experimental parameters of examples as shown in table 2, 10 × 4 to 40 examples were designed according to the number of workpieces and the cost factor of batch processing capacity, and 100 examples were randomly generated for each example, so there were 4000 examples in total.
TABLE 2 Experimental parameters
Figure GDA0001559085640000142
The simulation results are shown in fig. 1.
Examples illustrate that:
the invention is described below with reference to specific examples:
the information on the workpiece is as follows,
Figure GDA0001559085640000143
pmax＝12，m＝4，β＝0.3。
1) according to known conditions, can be obtained
Figure GDA0001559085640000144
So b2＝b1-1＝4。
2) Based on the batch processing capacity cost factor, the method can obtain
Figure GDA0001559085640000145
Therefore, it is not only easy to use
Figure GDA0001559085640000146
Figure GDA0001559085640000147
3) When b is more than or equal to b1When f is present1(b) Is a monotonically increasing function with a minimum value of minf1(b),b∈{b1,b3,b4}; when b is less than b1When f is present2(b) Is a monotonous decreasing function with a minimum value of f2(b2)。
4) Therefore, it is not only easy to use
Figure GDA0001559085640000148
Namely, it is
Figure GDA0001559085640000149
An example function graph is shown in fig. 3.
The batch processing capacity of the best economic benefit of the batch processor in the example is 4, and theoretical basis is provided for decisions such as old line modification and new line capacity planning of the batch processor.
Although embodiments of the present invention have been shown and described above, it is understood that the above embodiments are exemplary and should not be construed as limiting the present invention, and that variations, modifications, substitutions and alterations can be made in the above embodiments by those of ordinary skill in the art without departing from the principle and spirit of the present invention.

Claims (1)

1. An input-output decision method for considering the cost of a parallel batch processor is characterized by comprising the following steps: the method comprises the following steps:
step 1: establishing an input-output decision problem mathematical model of the capacity cost of the parallel batch processor:
the parallel batch processing machines are used for processing n workpieces, and the parallel batch processing machines can process b workpieces at most at the same time; the optimization objective is to minimize the maximum machining time CmaxSum of batch processor capacity cost:
Figure FDA0001559085630000018
beta is a batch processing capacity cost coefficient;
the batch scheduling problem is represented as: pm | p-batch, b < n | Cmax+βmb
The mathematical model is as follows:
obj.min(Cmax+βmb)
Figure FDA0001559085630000011
Figure FDA0001559085630000012
p(Bik)≥xjikpj j＝1,2,…,n，i＝1,2,…,m
C(Bik)≥C(Bi(k-1))+p(Bik) i＝1,2,…,m
Figure FDA0001559085630000013
xjik∈{0,1} i＝1,2,…,m，j＝1,2,…,n
wherein x isjikIs a variable of 0 to 1, when the workpiece JjAt machine MiIs processed and arranged at BkIn a batch, then xjik1, otherwise xjik＝0；
Step 2: loosening the workpiece to be interruptible to obtain a loosening problem of the original problem, and analyzing the optimal solution property of the loosening problem:
the batch scheduling problem Pm | p-batch established in step 1, b < n | CmaxAdding constraint condition that workpiece can be interrupted in + beta mb to obtain relaxation problem of batch scheduling problem Pm | p-batch, b < n, pmtn | Cmax+βmb；
For the relaxation problem Pm | p-batch, b < n, pmtn | CmaxCarrying out optimal solution on the beta mb to obtain the optimal solution
Figure FDA0001559085630000014
Wherein b is*Representing an objective function
Figure FDA0001559085630000015
Minimum batch processing capacity; s*(b*) Representing the optimal scheduling scheme when the batch processing capacity is b; cmax(S*(b*) Represents the maximum completion time under the optimal scheduling scheme;
a cost function of
Figure FDA0001559085630000016
Where the subscript p indicates that the workpiece can be interrupted,
Figure FDA0001559085630000017
when the batch processing capacity is b, the optimal scheduling scheme under the condition that the workpieces can be interrupted is represented, and beta represents a batch processing capacity cost coefficient; the cost function is expressed as:
Figure FDA0001559085630000021
wherein f is1(b)＝pmax+βmb，f2(b)＝P/mb+βmb；
And step 3: designing a heuristic algorithm 1 according to the optimal solution property of the relaxation problem, and solving the optimal batch processing capacity of the relaxation problem:
order to
Figure FDA0001559085630000022
Representing the optimal batch processing capacity when a workpiece can be interrupted, and representing the optimal cost function of the relaxation problem as
Figure FDA0001559085630000023
f1(b) Is a monotonically increasing function of batch processing capacity b, resulting in minf1(b)＝f1(b1) (ii) a Cost function f2(b) Is a convex function when
Figure FDA0001559085630000024
The time objective function obtains global optimum, when b is less than or equal to mu, f2(b) Is a monotone decreasing function, when b is more than or equal to mu, f2(b) Is a monotonically increasing function, obtaining
Figure FDA0001559085630000025
Thereby obtaining
Figure FDA0001559085630000026
To solve the problem of relaxation
Figure FDA0001559085630000027
Solving the batch processing capacity b for minimizing the cost function by the following steps1(ii) a Wherein the batch processing capacity b1To enable the maximum processing time of a workpiece to be not less than the minimum batch processing capacity at the average load of the machines for a given number of machines, p is satisfiedmax≥P/mb1Minimum batch processing capacity of
Figure FDA0001559085630000028
Wherein
Figure FDA0001559085630000029
Step 3.1: by the equation
Figure FDA00015590856300000210
Calculation of b1And make an order
Figure FDA00015590856300000211
Step 3.2: by the equation
Figure FDA00015590856300000212
Calculating mu;
step 3.2.1: if it is
Figure FDA00015590856300000213
Go to step 3.3, otherwise go to step 3.2;
step 3.2.2: let b2＝b11, if f2(b2)≤f1(b1) Then, then
Figure FDA00015590856300000214
Otherwise
Figure FDA00015590856300000215
Go to step 3.4;
step 3.3: computing
Figure FDA00015590856300000216
And
Figure FDA00015590856300000217
if f2(b4)≤f2(b3)≤f1(b1) Then, then
Figure FDA00015590856300000218
If f2(b3)≤f2(b4)≤f1(b1) Then, then
Figure FDA00015590856300000219
Otherwise
Figure FDA00015590856300000220
Step 3.4: output of
Figure FDA0001559085630000031
And
Figure FDA0001559085630000032
terminating;
and 4, step 4: designing a heuristic algorithm RPH, and solving a total cost function of the relaxation problem under the optimal batch processing capacity:
analyzing a scheduling problem considering batch processing capacity cost under the condition that workpieces cannot be interrupted
Pm|p-batch,b＜n|Cmax+βmb
The objective function after considering the cost of the batch processing capacity is
Figure FDA0001559085630000033
Where the subscript np indicates that the workpiece is not interruptible,
Figure FDA0001559085630000034
representing the optimal scheduling scheme under the condition that the workpieces cannot be interrupted when the batch processing capacity is b;
get
Figure FDA0001559085630000035
Represents the optimal batch processing capacity under the condition that workpieces cannot be interrupted, and has all batch processing capacities b more than or equal to 1
Figure FDA0001559085630000036
If true; under the condition of the same batch processing capacity, the upper bound of the ratio of the optimal target value when the workpiece can not be interrupted to the optimal target value when the workpiece can be interrupted is 2-1/mb;
the following heuristic algorithm is adopted to make a scheduling scheme:
step 4.1: obtaining the optimal batch processing capacity for the relaxation problem according to step 3
Figure FDA0001559085630000037
Order to
Figure FDA0001559085630000038
Step 4.2: according to b obtained in the step 4.1, the following heuristic algorithm H1 is operated, and a scheduling scheme C is outputmax(Snp(b) ) and cost function
Figure FDA0001559085630000039
Step 4.2.1: workpiece collection
Figure FDA00015590856300000310
All workpieces are batched according to the FBLPT rule to obtain k batches,
Figure FDA00015590856300000311
step 4.2.2: each batch of workpieces is scheduled on the machine that minimizes its completion time according to LPT rules until all batches are scheduled.
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