CN107609678A - A kind of homotype parallel machine production scheduling method for considering square information uncertainty - Google Patents
A kind of homotype parallel machine production scheduling method for considering square information uncertainty Download PDFInfo
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Abstract
The present invention proposes a kind of homotype parallel machine production scheduling method for considering square information uncertainty, belongs to production scheduling and operational research field.This method builds the distribution collection Robust Optimization Model DR PMSP MU for considering square information uncertainty first, obtains the expression formula for including the initial model DR PMSP MU1 of interior layer problems and outer layer problem；After being changed to the decision variables of DR PMSP MU models, to internal layer problem solving, and DR PMSP MU1 model equivalencies are converted into can solving model DR PMSP MU2；The optimal solution of DR PMSP MU2 models is the optimal solution of whole model, and the optimal solution corresponds to multiple Optimal Production scheduling schemes, and policymaker can voluntarily select as needed；The model that the present invention establishes more conforms to the situation of actual production, by using more information in production environment, in the case where ensureing systematic function, can reduce the risk of decision-making.
Description
Technical Field
The invention belongs to the field of production scheduling and operation research, and particularly relates to a homotype parallel machine production scheduling method considering moment information uncertainty.
Background
The scheduling problem of the parallel machines of the same type considers that a batch of workpieces are processed on a series of same machines, each machine can only process one workpiece at the same time, and each workpiece can only be processed by one machine at the same time. Each workpiece has specific processing time, the scheduling problem is to find a scheduling scheme, and all workpieces are distributed to corresponding machines for processing in a certain processing sequence, so that the performance index of a certain system is optimal. In an actual production manufacturing environment, there is often uncertainty in the processing time of a workpiece due to factors such as machine or tool conditions, worker processing level, and processing environment. And the trend of individual customization of small batches in the manufacturing industry is more and more obvious at present, and the processing time of some new products can only be estimated and obtained through some historical processing data, so that the uncertainty of the processing time of workpieces is continuously enhanced. With such a high degree of uncertainty, if the conventional deterministic model is still used to make scheduling decisions, it may lead to a severe deviation from the expectations of the decision-maker, and may even result in a scheduling scheme that is not feasible at all in practice. Therefore, researchers are beginning to gradually study the scheduling problem of parallel machines of the same type with uncertainty.
In the research of scheduling problems of parallel machines of the same type with uncertainty, two methods are mainly used at present, wherein one method is to establish a random scheduling model, and the other method is to establish a robust scheduling model. In the stochastic scheduling model, uncertain workpiece processing time is regarded as a random variable with known distribution, and the optimization target of the model is often the long-term performance expectation of the system. Although the stochastic scheduling model has a good propulsion function on the research of the uncertain parallel machine scheduling problem in theory, some inherent defects of the stochastic scheduling model limit the application of the stochastic scheduling model in the practical large-scale scheduling problem. These disadvantages are mainly manifested in the following aspects: 1) in the stochastic scheduling model, the distribution of uncertain parameters needs to be known accurately. However, in an actual production environment, as the production operation becomes more complicated and the production modes such as small-lot personalized customization gradually change, in many cases, an accurate probability distribution is difficult to obtain, and the interval range of the product can only be estimated according to the processing time of similar products. This problem is particularly acute in connection with new products, especially single-piece or single-process products. In this case, the stochastic scheduling model will no longer apply. 2) In the stochastic scheduling model, some expectation of system performance is typically employed as an optimization objective. Such goals are more suitable for planning the development of an enterprise over a long period of time and are not suitable for solving the problem of maximizing benefits or minimizing risks during each actual operation. 3) Such stochastic models are generally NP-hard, and can generally be solved only by heuristic algorithms or dynamic programming algorithms, and as the scale of the problem gradually expands, the difficulty in solving the stochastic scheduling model will increase exponentially.
Another robust scheduling model was originally proposed by Richard l.daniels et al, in which the workpiece processing time, only of which the interval information is known, is characterized by interval data scenarios (one scenario represents one possible value of the workpiece processing time), which is simpler and more practical than the description of the parameter distribution function in the stochastic scheduling model. The current robust parallel machine scheduling problem is to adopt a robust optimization method based on an uncertainty set, wherein the uncertainty set is a limited discrete set or a continuous interval form. In this model, the key problem of robust scheduling is how to define the worst environment, find the robust cost of each feasible solution in the worst environment, and find the optimum among the robust costs of all feasible solutions. The robust scheduling model based on the uncertainty set is more suitable for the parameter situation of actual production, and a robust decision with better system performance under the worst condition can be found, so that the risk of the decision is reduced. However, because only the boundary information of the workpiece processing time variation range is utilized and the system performance under the worst condition is mainly considered, the decision obtained by the robust scheduling model based on the uncertainty set is possibly too conservative, and the system performance under the normal condition of the parameters is sacrificed. Therefore, how to utilize more information of the historical data and reduce the conservative degree of the decision while ensuring the robustness is an urgent problem to be solved in the current robust scheduling.
In the optimization problem where the process parameters have uncertainty, there is a new optimization method called distribution set robust optimization. Although the method is rarely applied to the field of production scheduling at present, the method is widely applied to a plurality of fields of electric power, health, securities and the like, and achieves good effect. In the distribution set robust optimization method, an uncertain parameter is represented by a random variable, but the distribution function of the random variable is unknown and belongs to a certain distribution function set. In the optimization process, all possible distribution functions in the set of distribution functions need to be considered. Although the uncertain parameters are still regarded as a random variable in the distribution set robust optimization model, compared with the random optimization model, the specific form of the distribution function is not required to be determined, and only a set of the distribution function is determined. Compared with the robust optimization model based on the uncertainty set, the robust optimization of the distribution set not only utilizes the information of the parameter variation range, but also takes more information such as the mean value, the variance and the like into consideration so as to reduce the conservative degree of decision.
Disclosure of Invention
The invention aims to overcome the defects of the prior art and provides a method for producing and scheduling a parallel machine of the same type by considering moment information uncertainty. The model established by the invention better accords with the actual production condition, and by utilizing more information in the production environment, the decision risk can be reduced under the condition of ensuring the system performance, and the optimal production scheduling scheme can be obtained.
The invention provides a homotype parallel machine production scheduling method considering moment information uncertainty, which is characterized by comprising the following steps of:
1) aiming at the scheduling problem of a same-type parallel machine, a distribution set robust optimization model DR-PMSP-MU considering moment information uncertainty is constructed, and an expression of an initial model DR-PMSP-MU1 is obtained;
in a DR-PMSP-MU model, the performance index of the system is selected as total flow passing time TFT; assuming that all workpieces are released at the moment when the machining starts, namely the release time is 0, the machining time of the workpieces has random uncertainty, the distribution of the random machining time is unknown, but belongs to a distribution set determined by a support set, a sampling mean vector and a sampling covariance matrix; the DR-PMSP-MU model aims to find an optimal robust scheduling scheme, so that the scheduling scheme has the minimum expected TFT under the condition that the workpiece processing time obeys worst distribution;
1-1) determining model decision variables;
decision variables of the DR-PMSP-MU model are feasible scheduling schemes, and include machine selection of all workpieces and processing sequences on each machine; given that there are J artifacts and M machines in the model, the set of artifacts and machines are J ═ {1, 2., J } and M ═ 1, 2., M } respectively, then a feasible scheduling scheme is formed by a three-dimensional matrix X ∈ {0,1}J×M×J＝{xjmlBelongs to {0,1} | J belongs to J, M belongs to M, L belongs to L and J }; wherein if a workpiece j is assigned to the m-th machine and processed in order of the last but one, xjml1, otherwise xjml＝0；
1-2) random vector representation of processing time;
the processing time of all the workpieces is a random vector p, the distribution obeyed by the vector is unknown, but belongs to a distribution set determined by a support set, a sampling mean vector and a sampling covariance matrix, and the expression of the distribution set is shown as formula (2):
wherein,indicates that the machining time of each workpiece is non-negative, Ep]True mean value representing all workpiece processing timesVector, musAnd sigmasRespectively representing the sampling mean vector and the sampling covariance matrix of all the workpiece processing time, wherein gamma is a constraint parameter;
1-3) determining an objective function of a DR-PMSP-MU model;
given a schedule X and all workpiece processing time vectors p, the TFT is calculated by equation (3):
in the formula, pjRepresents the processing time of the workpiece j;
the total flow passing time TFT is a random variable, the expectation is taken as the measurement of the random TFT, the objective function E [ F (X, p) ] of the DR-PMSP-MU model is obtained, and the total flow passing time expectation value obtained when all the workpiece processing time vectors p obey a certain distribution function F is represented;
1-4) determining a constraint condition of a DR-PMSP-MU model;
1-4-1) random processing time constraint;
the distribution of the processing time vectors p of all the workpieces is unknown, but belongs to a distribution set determined by a support set, a sampling mean vector and a sampling covariance matrix, and the expression is shown as formula (4):
1-4-2) feasible scheduling scheme constraints;
each element in the feasible scheduling scheme X is a variable from 0 to 1, and the expression is shown as formula (5):
1-4-3) workpiece occupation position constraint;
each workpiece only occupies one position on one machine, and the expression is shown in the formula (6):
1-4-4) the position is occupied and restricted by the workpiece;
each position on each machine is occupied by at most one workpiece, and the expression is shown in the formula (7):
1-4-5) ordering compact constraints;
the occupied positions on each machine are consecutive and starting from 1, the expression is given by the equation (8):
the latter four kinds of constraints shown in equations (5) to (8) are all feasible for restricting the scheduling scheme, and are integrated together to form a feasible domain X of the scheduling scheme, as shown in equation (9):
1-5) establishing an expression of a same-type parallel machine scheduling distribution set robust initial model DR-PMSP-MU1 considering moment information uncertainty, as shown in formula (10):
in the formula, max is represented in the distribution set DmIn search results in an objective function E [ f (X, p)]The distribution function F, min with the maximum value represents that the inner layer problem is searched in the feasible field X of the scheduling schemeThe scheduling scheme X with the minimum optimal value;
2) performing equivalent transformation on the DR-PMSP-MU1 model established in the step 1);
2-1) converting decision variables;
equivalently converting decision variables of the DR-PMSP-MU model from a three-dimensional matrix X into a two-dimensional matrix Y, wherein the conversion relation is shown as a formula (11):
the expression of the feasible domain of Y is shown as formula (12):
the two-dimensional matrix Y is expressed as a vector pi, and the expression is shown as the formula (13):
pi represents the reverse order of the processing sequence of the workpiece after the machine serial number is ignored;
the expression of the feasible domain of pi is shown as the formula (14):
the TFT is expressed as the inner product of pi and p, and the expression is shown as the formula (15):
f(π,p)＝f(X,p)＝πTp； (15)
2-2) solving an inner layer problem;
after the decision variable X is converted into pi in the step 2-1), according to E [ f (pi, p)]＝πTE[p]The inner layer problem in the DR-PMSP-MU1 model shown in equation (10) is simplified as shown in equation (16):
wherein, mu-E [ p ] is a real mean vector of all the workpiece processing time, and s.t. represents a constraint condition;
solving equation (16) to obtain the optimal solution ofThe optimal value of the inner layer problem in the corresponding DR-PMSP-MU1 model is
2-3) equivalently converting the DR-PMSP-MU1 model into a solvable model DR-PMSP-MU 2;
replacing the inner layer problem in the formula (10) with the optimal value obtained in the step 2-2), equivalently converting the DR-PMSP-MU1 model into a DR-PMSP-MU2 model, wherein the expression of the DR-PMSP-MU2 model is shown in the formula (17):
3) solving the DR-PMSP-MU model to obtain an optimal production scheduling scheme;
solving the DR-PMSP-MU2 model shown in the formula (17), wherein the obtained optimal solution of the DR-PMSP-MU2 model is the optimal solution of the DR-PMSP-MU model; the optimal solution is an optimal vector pi value, and the vector pi value is equivalently converted into an optimal two-dimensional matrix Y value containing the processing sequence corresponding to each workpiece through the corresponding relation shown in the formula (13), so that an optimal production scheduling scheme is obtained.
The invention has the characteristics and beneficial effects that: :
1) according to the homotype parallel machine production scheduling method considering moment information uncertainty, a distribution set robust model is established on the basis of a traditional certainty model, only a support set, a sampling mean value and a sampling covariance matrix of random processing time are needed, distribution information of the distribution set robust model does not need to be accurately known, and compared with a random model, the homotype parallel machine production scheduling method is more suitable for actual production and has higher practicability.
2) Compared with a robust scheduling model only using an interval change range, the distribution set robust model adopted by the invention has smaller conservative property. By utilizing more information, the obtained optimal robust solution has better system performance on the premise of ensuring the robustness.
3) Since both the mean and the variance of the workpiece processing time are obtained by sampling the historical data in actual production, the obtained sampled data may be biased in the case where the available historical data is insufficient. The invention fully considers the situation, and brings the uncertainty of the sampling data into the model, and the designed homotype parallel machine scheduling method considering the uncertainty of the moment information meets the actual requirement.
4) The model established by the invention can control the trust degree of the sampled data through parameter setting, so that a decision maker can set corresponding parameter values according to the current requirements to obtain the scheduling strategy most suitable for the current production situation.
5) The distributed set robust optimization model adopted by the invention can be combined with the advantages of a random scheduling model and a robust scheduling model based on an uncertainty set, is applied to the production scheduling problem of the parallel machines of the same type, better accords with the actual production condition compared with the existing robust modeling method, and can reduce the risk of decision by utilizing more information in the production environment under the condition of ensuring the system performance.
Detailed Description
The invention provides a homotype parallel machine production scheduling method considering moment information uncertainty, which is further described in detail below by combining specific embodiments.
The invention provides a homotype parallel machine production scheduling method considering moment information uncertainty, which comprises the following steps:
1) aiming at the scheduling problem of a same-type parallel machine, a distribution set robust optimization model DR-PMSP-MU considering moment information uncertainty is constructed, and an expression of an initial model DR-PMSP-MU1 is obtained;
the invention relates to a scheduling problem of a same-type parallel machine with uncertain processing time, and a distributed set robust optimization model (DR-PMSP-MU) considering moment information uncertainty is established aiming at the scheduling problem. In the same type of parallel machine scheduling problem, all workpieces can be processed on any one machine, but each machine can process only one workpiece at the same time, and one workpiece can be allocated to only one machine. Each workpiece has its own specific machining time and cannot be interrupted during machining. The purpose of solving the scheduling problem of the parallel machines of the same type is to find a scheduling scheme, determine the machine allocation conditions of all the workpieces and the processing sequence on each machine, and enable one of the system performance indexes such as total flow time, maximum delay time or delay workpiece number to be optimal.
In the DR-PMSP-MU model, the performance index of the system is selected as Total Flow Time (TFT), and it is assumed that all workpieces are released at the time when machining starts, i.e., the release time is 0. The processing time of the workpiece has random uncertainty, and the distribution of the random processing time is unknown but belongs to a distribution set determined by a support set, a sampling mean vector and a sampling covariance matrix. Under this setting, the goal of the DR-PMSP-MU model is to find an optimal robust scheduling scheme that has the smallest expected TFT if the workpiece processing time follows the worst distribution.
1-1) determining model decision variables;
decision variables of the DR-PMSP-MU model are feasible scheduling schemes, including machine selection of all workpieces and machining order on each machine. Given that there are J artifacts and M machines in the model, the set of artifacts and machines are J ═ {1, 2., J } and M ═ 1, 2., M } respectively, then a feasible scheduling scheme is formed by a three-dimensional matrix X ∈ {0,1}J×M×J＝{xjmlBelongs to {0,1} | J belongs to J, M belongs to M, L belongs to L and J }; wherein if a workpiece j is assigned to the m-th machine and processed in order of the last but one, xjml1, otherwise xjml＝0。
For example, for a parallel machine scheduling problem with 5 workpieces, 2 machines, a feasible scheduling scheme SkCan be written as:
the machine 1: workpiece 1-workpiece 2-workpiece 4; the machine 2: workpiece 3-workpiece 5.
Scheduling scheme S according to the corresponding relation between the scheduling scheme and XkThree-dimensional matrix X ofkThe expression is shown in formula (1):
in the formula, Xk(m-1) and Xk(m ═ 2) respectively denotes a two-dimensional matrix corresponding to the machine 1 and a two-dimensional matrix corresponding to the machine 2; these two-dimensional matrices together form a scheduling scheme S representing a feasible solutionkThree-dimensional matrix X ofk。
1-2) random vector representation of processing time;
the DR-PMSP-MU model of the invention regards the processing time of all workpieces as a random vector p. The distribution obeyed by the method is unknown, but belongs to a distribution set determined by a support set, a sampling mean vector and a sampling covariance matrix, and the expression of the distribution set is shown as a formula (2):
wherein,indicating that the machining time for each workpiece is non-negative, E p]True mean vector, mu, representing the processing time of a workpiecesAnd sigmasAnd the sampling mean vector and the sampling covariance matrix respectively represent the processing time of the workpiece, and gamma is a constraint parameter and reflects the trust degree of the sampling information.
1-3) determining an objective function of a DR-PMSP-MU model;
the system performance index of the DR-PMSP-MU model is Total Flow Time (TFT), and when a scheduling scheme X and all workpiece processing time vectors p are given, the TFT is calculated by the formula (3):
in the formula, pjThe machining time of the workpiece j is shown.
Because the processing time p of all the workpieces is a random vector, and the total flow time TFT is a random variable, the method adopts expectation as the measurement of the random TFT, obtains the objective function E [ F (X, p) ] of the DR-PMSP-MU model, and expresses the expected value of the total flow time obtained when the processing time vectors p of all the workpieces obey a certain distribution function F.
1-4) determining a constraint condition of a DR-PMSP-MU model;
the DR-PMSP-MU model contains 5 constraint conditions, of which 1 is a distribution set to which the constraint random machining time obeys, and the other 4 is feasibility of a constraint scheduling scheme, as shown in detail below:
1-4-1) random processing time constraint;
the distribution of the processing time vectors p of all the workpieces is unknown, but belongs to a distribution set determined by a support set, a sampling mean vector and a sampling covariance matrix, and the expression is shown as formula (4):
1-4-2) feasible scheduling scheme constraints;
each element in the feasible scheduling scheme X is a variable from 0 to 1, and the expression is shown as formula (5):
1-4-3) workpiece occupation position constraint;
each workpiece can only occupy one position on one machine, and the expression is shown in the formula (6):
1-4-4) the position is occupied and restricted by the workpiece;
each position on each machine can be occupied by at most one workpiece, and the expression is shown in the formula (7):
1-4-5) ordering compact constraints;
the occupied positions on each machine must be consecutive and starting from 1, the expression is given by the equation (8):
the latter four kinds of constraints shown in equations (5) to (8) are all feasible for restricting the scheduling scheme, and are integrated together to form a feasible domain X of the scheduling scheme, as shown in equation (9):
1-5) establishing an expression of a same-type parallel machine scheduling distribution set robust initial model DR-PMSP-MU1 considering moment information uncertainty, as shown in formula (10):
in the formula, max is represented in the distribution set DmIn search results in an objective function E [ f (X, p)]The distribution function F, min with the maximum value represents that the inner layer problem is searched in the feasible field X of the scheduling schemeThe scheduling scheme X having the smallest optimal value.
2) Performing equivalent transformation on the DR-PMSP-MU1 model established in the step 1);
2-1) converting decision variables;
because the machine serial number allocated to the workpiece has no influence on the objective function value, the decision variables of the DR-PMSP-MU model can be equivalently converted into a two-dimensional matrix Y from a three-dimensional matrix X, and the conversion relation is shown as a formula (11):
the feasible fields for Y are written directly from X, and the expression is shown in formula (12):
to further facilitate the representation and calculation of the model, the two-dimensional matrix Y is represented as a vector pi, and the expression is shown in formula (13):
pi denotes the reverse order of the order in which the workpieces are machined after ignoring the machine number, e.g. pijI denotes the last but one sequence of machining of the workpiece j on a certain machine. The expression of the feasible domain of pi is shown as the formula (14):
the TFT can be expressed as the inner product of pi and p, and the expression is shown as the formula (15):
f(π,p)＝f(X,p)＝πTp； (15)
2-2) solving an inner layer problem;
after converting the decision variable X into pi in step 2-1), attention is paid to E [ f (pi, p)]＝πTE[p]The inner-layer maximization problem in the DR-PMSP-MU1 model shown in equation (10) can be simplified as shown in equation (16):
where μ ═ E [ p ] is the true mean vector of all workpiece processing times, and s.t. represents the constraint.
Since the maximization problem shown in equation (16) is a convex optimization problem with a linear objective function, the optimal solution thereof must exist. We find the optimal solution of the problem through Karush-Kuhn-Tucke (KKT) requirements asThe optimal value of the inner layer problem in the corresponding DR-PMSP-MU1 model is
2-3) equivalently converting the DR-PMSP-MU1 model into a solvable model DR-PMSP-MU 2;
replacing the inner layer maximization problem in the formula (10) with the optimal value obtained in the step 2-2), equivalently converting the DR-PMSP-MU1 model into a DR-PMSP-MU2 model, wherein the expression of the DR-PMSP-MU2 model is shown in the formula (17):
3) solving the DR-PMSP-MU model to obtain an optimal production scheduling scheme;
the DR-PMSP-MU2 model shown in formula (17) is an integer second-order cone programming model, and can be solved through IBM CPLEX and other commercial solvers, and can also be solved through designing a more efficient iterative descent algorithm according to the nature of the problem.
The obtained optimal solution of the DR-PMSP-MU2 model is the optimal solution of the DR-PMSP-MU model, the optimal solution is an optimal vector pi value, and the optimal solution can be equivalently converted back to an optimal two-dimensional matrix Y through the corresponding relation shown in the formula (13), wherein the optimal solution comprises the processing sequence information corresponding to each workpiece; the optimal two-dimensional matrix Y value can correspond to a plurality of optimal production scheduling schemes, and a decision maker can select the optimal two-dimensional matrix Y value according to needs in actual production.
For example, for a workpiece having 5 workpieces,the scheduling problem of the parallel machines of the same type of 2 machines is solved, if the optimal vector pi value obtained by the DR-PMSP-MU model is pi ═ 1, 3, 2, 2, 1)TThen, the corresponding Y value of the two-dimensional matrix is as shown in equation (18):
in the formula, the row number indicates the serial number of the workpiece, the column number indicates the processing order of the reciprocal, and for example, the element "1" on the third column of the second row indicates that the workpiece 2 is processed at the third position of the reciprocal on a certain machine.
Because all machines are the same and the processing effects are equivalent in parallel machines of the same type, when the optimal scheduling scheme is determined, the processing sequence of the workpieces only needs to be ensured to meet the requirements of the optimal solution pi and Y, and for the serial numbers of the machines, the serial numbers of the machines can be arbitrarily selected on the premise of ensuring that each machine only processes one workpiece at the same time.
The optimal two-dimensional matrix Y values shown in equation (18) may correspond to multiple optimal scheduling schemes, such as shown in equations (19) -21:
scheme 1. machine 1: workpiece 2-workpiece 3-workpiece 1; the machine 2: workpiece 4-workpiece 5; (19)
scheme 2. machine 1: workpiece 2-workpiece 4-workpiece 5; the machine 2: workpiece 3 — workpiece 1; (20)
scheme 3. machine 1: workpiece 3 — workpiece 1; the machine 2: workpiece 2-workpiece 4-workpiece 5; (21)
although the scheduling schemes are different in machine allocation, the obtained DR-PMSP-MU models have the same objective function value and belong to equivalent scheduling schemes. In actual production, a decision maker can select the materials according to needs.
In a specific embodiment of the present invention, a robust scheduling scheme obtained by a same-type parallel machine production scheduling method considering moment information uncertainty is compared with a deterministic scheduling scheme obtained only considering mean information, so as to demonstrate that the same-type parallel machine scheduling method considering moment information uncertainty designed by the present invention can greatly reduce interference caused by sampling errors, improve the robustness of a system, and the specific analysis is as follows:
the value of gamma reflects the trust degree of the sampled information and can be obtained by analyzing historical data. When historical data of a batch of workpiece processing time is obtained, the data is divided into two parts on average. One of which is used to calculate the sample mean vector musWith the sampling covariance matrix ∑s(ii) a The other for a given gamma value, such that the mean value of this part of the data E p]Located on an ellipsoidAmong them.
In this example, the true mean of the random processing time is in the interval [10,60 ]]Is arbitrarily selected, and the variance is influenced by the mean value and is selected asFor each pair of mean and variance examples, 10,000 processing time examples were randomly generated as historical data and controlled by sampling rate for calculating μsAnd sigmasThe ratio of sampled data. Table 1 shows the robustness improvement and mean loss obtained for the robust scheduling scheme under different sampling rate values.
Table 1 TFT statistical comparison table for robust scheduling scheme and deterministic scheduling scheme under different sampling rate values
The results in table 1 show that:
1) under the same sampling rate, compared with a deterministic scheduling scheme, the robust scheduling scheme obtained by the method of the invention slightly increases the mean value of the TFT and obviously decreases the variance. Namely, the distribution set robust model considering moment information uncertainty is at the cost of small average performance loss, so that the dispersity of the TFT is reduced, the deviation of sampling data can be effectively resisted, and the robustness of a production system is improved.
2) The higher sampling rate causes the value of gamma to become smaller and the two scheduling schemes behave more closely. This shows that the advantage of robust scheduling decision is more significant in the case of more serious historical data loss, i.e. greater deviation of sampling information. The important significance of the homotype parallel machine scheduling method considering moment information uncertainty in the actual production environment with high uncertainty and less historical processing data is fully embodied.
Claims (1)
1. A homotype parallel machine production scheduling method considering moment information uncertainty is characterized by comprising the following steps:
1) aiming at the scheduling problem of a same-type parallel machine, a distribution set robust optimization model DR-PMSP-MU considering moment information uncertainty is constructed, and an expression of an initial model DR-PMSP-MU1 is obtained;
in a DR-PMSP-MU model, the performance index of the system is selected as total flow passing time TFT; assuming that all workpieces are released at the moment when the machining starts, namely the release time is 0, the machining time of the workpieces has random uncertainty, the distribution of the random machining time is unknown, but belongs to a distribution set determined by a support set, a sampling mean vector and a sampling covariance matrix; the DR-PMSP-MU model aims to find an optimal robust scheduling scheme, so that the scheduling scheme has the minimum expected TFT under the condition that the workpiece processing time obeys worst distribution;
1-1) determining model decision variables;
decision variables of the DR-PMSP-MU model are feasible scheduling schemes, and include machine selection of all workpieces and processing sequences on each machine; given that there are J artifacts and M machines in the model, the set of artifacts and machines are J ═ {1, 2., J } and M ═ 1, 2., M } respectively, then a feasible scheduling scheme is formed by a three-dimensional matrix X ∈ {0,1}J×M×J＝{xjmlBelongs to {0,1} | J belongs to J, M belongs to M, L belongs to L and J }; wherein if a workpiece j is assigned to the m-th machine and processed in order of the last but one, xjml1, otherwise xjml＝0；
1-2) random vector representation of processing time;
the processing time of all the workpieces is a random vector p, the distribution obeyed by the vector is unknown, but belongs to a distribution set determined by a support set, a sampling mean vector and a sampling covariance matrix, and the expression of the distribution set is shown as formula (2):
wherein,indicates that the machining time of each workpiece is non-negative, Ep]True mean vector, μ, representing the processing time of all workpiecessAnd sigmasRespectively representing the sampling mean vector and the sampling covariance matrix of all the workpiece processing time, wherein gamma is a constraint parameter;
1-3) determining an objective function of a DR-PMSP-MU model;
given a schedule X and all workpiece processing time vectors p, the TFT is calculated by equation (3):
<mrow> <mi>f</mi> <mrow> <mo>(</mo> <mi>X</mi> <mo>,</mo> <mi>p</mi> <mo>)</mo> </mrow> <mo>=</mo> <munderover> <mo>&Sigma;</mo> <mrow> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>J</mi> </munderover> <munderover> <mo>&Sigma;</mo> <mrow> <mi>m</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>M</mi> </munderover> <munderover> <mo>&Sigma;</mo> <mrow> <mi>l</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>J</mi> </munderover> <msub> <mi>lp</mi> <mi>j</mi> </msub> <msub> <mi>x</mi> <mrow> <mi>j</mi> <mi>m</mi> <mi>l</mi> </mrow> </msub> <mo>,</mo> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>3</mn> <mo>)</mo> </mrow> </mrow>
in the formula, pjRepresents the processing time of the workpiece j;
the total flow passing time TFT is a random variable, the expectation is taken as the measurement of the random TFT, the objective function E [ F (X, p) ] of the DR-PMSP-MU model is obtained, and the total flow passing time expectation value obtained when all the workpiece processing time vectors p obey a certain distribution function F is represented;
1-4) determining a constraint condition of a DR-PMSP-MU model;
1-4-1) random processing time constraint;
the distribution of the processing time vectors p of all the workpieces is unknown, but belongs to a distribution set determined by a support set, a sampling mean vector and a sampling covariance matrix, and the expression is shown as formula (4):
1-4-2) feasible scheduling scheme constraints;
each element in the feasible scheduling scheme X is a variable from 0 to 1, and the expression is shown as formula (5):
<mrow> <msub> <mi>x</mi> <mrow> <mi>j</mi> <mi>m</mi> <mi>l</mi> </mrow> </msub> <mo>&Element;</mo> <mo>{</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>}</mo> <mo>,</mo> <mo>&ForAll;</mo> <mi>j</mi> <mo>&Element;</mo> <mi>J</mi> <mo>,</mo> <mo>&ForAll;</mo> <mi>m</mi> <mo>&Element;</mo> <mi>M</mi> <mo>,</mo> <mo>&ForAll;</mo> <mi>l</mi> <mo>&Element;</mo> <mi>L</mi> <mo>;</mo> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>5</mn> <mo>)</mo> </mrow> </mrow>
1-4-3) workpiece occupation position constraint;
each workpiece only occupies one position on one machine, and the expression is shown in the formula (6):
<mrow> <munderover> <mo>&Sigma;</mo> <mrow> <mi>m</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>M</mi> </munderover> <munderover> <mo>&Sigma;</mo> <mrow> <mi>l</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>J</mi> </munderover> <msub> <mi>x</mi> <mrow> <mi>j</mi> <mi>m</mi> <mi>l</mi> </mrow> </msub> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo>&ForAll;</mo> <mi>j</mi> <mo>&Element;</mo> <mi>J</mi> <mo>;</mo> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>6</mn> <mo>)</mo> </mrow> </mrow>
1-4-4) the position is occupied and restricted by the workpiece;
each position on each machine is occupied by at most one workpiece, and the expression is shown in the formula (7):
<mrow> <munderover> <mo>&Sigma;</mo> <mrow> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>J</mi> </munderover> <msub> <mi>x</mi> <mrow> <mi>j</mi> <mi>m</mi> <mi>l</mi> </mrow> </msub> <mo>&le;</mo> <mn>1</mn> <mo>,</mo> <mo>&ForAll;</mo> <mi>m</mi> <mo>&Element;</mo> <mi>M</mi> <mo>,</mo> <mo>&ForAll;</mo> <mi>l</mi> <mo>&Element;</mo> <mi>L</mi> <mo>;</mo> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>7</mn> <mo>)</mo> </mrow> </mrow>
1-4-5) ordering compact constraints;
the occupied positions on each machine are consecutive and starting from 1, the expression is given by the equation (8):
<mrow> <munderover> <mo>&Sigma;</mo> <mrow> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>J</mi> </munderover> <msub> <mi>x</mi> <mrow> <mi>j</mi> <mi>m</mi> <mi>l</mi> </mrow> </msub> <mo>&GreaterEqual;</mo> <munderover> <mo>&Sigma;</mo> <mrow> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>J</mi> </munderover> <msub> <mi>x</mi> <mrow> <mi>j</mi> <mi>m</mi> <mrow> <mo>(</mo> <mi>l</mi> <mo>+</mo> <mn>1</mn> <mo>)</mo> </mrow> </mrow> </msub> <mo>,</mo> <mo>&ForAll;</mo> <mi>m</mi> <mo>&Element;</mo> <mi>M</mi> <mo>,</mo> <mo>&ForAll;</mo> <mi>l</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo>...</mo> <mo>,</mo> <mi>J</mi> <mo>-</mo> <mn>1</mn> <mo>;</mo> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>8</mn> <mo>)</mo> </mrow> </mrow>
the latter four kinds of constraints shown in equations (5) to (8) are all feasible for restricting the scheduling scheme, and are integrated together to form a feasible domain X of the scheduling scheme, as shown in equation (9):
<mrow> <mtable> <mtr> <mtd> <mrow> <mi>X</mi> <mo>=</mo> <mo>{</mo> <mi>X</mi> <mo>&Element;</mo> <msup> <mrow> <mo>{</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>}</mo> </mrow> <mrow> <mi>J</mi> <mo>&times;</mo> <mi>M</mi> <mo>&times;</mo> <mi>J</mi> </mrow> </msup> <mo>|</mo> <munderover> <mo>&Sigma;</mo> <mrow> <mi>m</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>M</mi> </munderover> <munderover> <mo>&Sigma;</mo> <mrow> <mi>l</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>J</mi> </munderover> <msub> <mi>x</mi> <mrow> <mi>j</mi> <mi>m</mi> <mi>l</mi> </mrow> </msub> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo>&ForAll;</mo> <mi>j</mi> <mo>&Element;</mo> <mi>J</mi> <mo>;</mo> <munderover> <mo>&Sigma;</mo> <mrow> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>J</mi> </munderover> <msub> <mi>x</mi> <mrow> <mi>j</mi> <mi>m</mi> <mi>l</mi> </mrow> </msub> <mo>&le;</mo> <mn>1</mn> <mo>,</mo> <mo>&ForAll;</mo> <mi>m</mi> <mo>&Element;</mo> <mi>M</mi> <mo>,</mo> <mo>&ForAll;</mo> <mi>l</mi> <mo>&Element;</mo> <mi>L</mi> <mo>;</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <munderover> <mo>&Sigma;</mo> <mrow> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>J</mi> </munderover> <msub> <mi>x</mi> <mrow> <mi>j</mi> <mi>m</mi> <mi>l</mi> </mrow> </msub> <mo>&GreaterEqual;</mo> <munderover> <mo>&Sigma;</mo> <mrow> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>J</mi> </munderover> <msub> <mi>x</mi> <mrow> <mi>j</mi> <mi>m</mi> <mrow> <mo>(</mo> <mi>l</mi> <mo>+</mo> <mn>1</mn> <mo>)</mo> </mrow> </mrow> </msub> <mo>,</mo> <mo>&ForAll;</mo> <mi>m</mi> <mo>&Element;</mo> <mi>M</mi> <mo>,</mo> <mo>&ForAll;</mo> <mi>l</mi> <mo>&Element;</mo> <mn>1</mn> <mo>,</mo> <mn>...</mn> <mo>,</mo> <mi>J</mi> <mo>-</mo> <mn>1</mn> <mo>}</mo> </mrow> </mtd> </mtr> </mtable> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>9</mn> <mo>)</mo> </mrow> </mrow>
1-5) establishing an expression of a same-type parallel machine scheduling distribution set robust initial model DR-PMSP-MU1 considering moment information uncertainty, as shown in formula (10):
<mrow> <munder> <mrow> <mi>m</mi> <mi>i</mi> <mi>n</mi> </mrow> <mrow> <mi>X</mi> <mo>&Element;</mo> <mi>X</mi> </mrow> </munder> <munder> <mi>max</mi> <mrow> <mi>F</mi> <mo>&Element;</mo> <msup> <mi>D</mi> <mi>m</mi> </msup> </mrow> </munder> <mi>E</mi> <mo>&lsqb;</mo> <mi>f</mi> <mrow> <mo>(</mo> <mi>X</mi> <mo>,</mo> <mi>p</mi> <mo>)</mo> </mrow> <mo>&rsqb;</mo> <mo>,</mo> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>10</mn> <mo>)</mo> </mrow> </mrow>
in the formula, max is represented in the distribution set DmIn search results in an objective function E [ f (X, p)]The distribution function F, min with the maximum value represents that the inner layer problem is searched in the feasible field X of the scheduling schemeThe scheduling scheme X with the minimum optimal value;
2) performing equivalent transformation on the DR-PMSP-MU1 model established in the step 1);
2-1) converting decision variables;
equivalently converting decision variables of the DR-PMSP-MU model from a three-dimensional matrix X into a two-dimensional matrix Y, wherein the conversion relation is shown as a formula (11):
<mrow> <msub> <mi>y</mi> <mrow> <mi>j</mi> <mi>l</mi> </mrow> </msub> <mo>=</mo> <munderover> <mo>&Sigma;</mo> <mrow> <mi>m</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>M</mi> </munderover> <msub> <mi>x</mi> <mrow> <mi>j</mi> <mi>m</mi> <mi>l</mi> </mrow> </msub> <mo>,</mo> <mo>&ForAll;</mo> <mi>j</mi> <mo>&Element;</mo> <mi>J</mi> <mo>,</mo> <mo>&ForAll;</mo> <mi>l</mi> <mo>&Element;</mo> <mi>L</mi> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>11</mn> <mo>)</mo> </mrow> </mrow>
the expression of the feasible domain of Y is shown as formula (12):
<mrow> <mi>Y</mi> <mo>=</mo> <mo>{</mo> <mi>Y</mi> <mo>&Element;</mo> <msup> <mrow> <mo>{</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>}</mo> </mrow> <mrow> <mi>J</mi> <mo>&times;</mo> <mi>J</mi> </mrow> </msup> <mo>|</mo> <munderover> <mo>&Sigma;</mo> <mrow> <mi>l</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>J</mi> </munderover> <msub> <mi>y</mi> <mrow> <mi>j</mi> <mi>l</mi> </mrow> </msub> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo>&ForAll;</mo> <mi>j</mi> <mo>&Element;</mo> <mi>J</mi> <mo>;</mo> <munderover> <mo>&Sigma;</mo> <mrow> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>J</mi> </munderover> <msub> <mi>y</mi> <mrow> <mi>j</mi> <mi>l</mi> </mrow> </msub> <mo>&le;</mo> <mi>M</mi> <mo>,</mo> <mo>&ForAll;</mo> <mi>l</mi> <mo>&Element;</mo> <mi>L</mi> <mo>;</mo> <munderover> <mo>&Sigma;</mo> <mrow> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>J</mi> </munderover> <msub> <mi>y</mi> <mrow> <mi>j</mi> <mi>l</mi> </mrow> </msub> <mo>&GreaterEqual;</mo> <munderover> <mo>&Sigma;</mo> <mrow> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>J</mi> </munderover> <msub> <mi>y</mi> <mrow> <mi>j</mi> <mrow> <mo>(</mo> <mi>l</mi> <mo>+</mo> <mn>1</mn> <mo>)</mo> </mrow> </mrow> </msub> <mo>,</mo> <mo>&ForAll;</mo> <mi>l</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>...</mn> <mo>,</mo> <mi>J</mi> <mo>-</mo> <mn>1</mn> <mo>}</mo> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>12</mn> <mo>)</mo> </mrow> </mrow>
the two-dimensional matrix Y is expressed as a vector pi, and the expression is shown as the formula (13):
<mrow> <msub> <mi>&pi;</mi> <mi>j</mi> </msub> <mo>=</mo> <munderover> <mo>&Sigma;</mo> <mrow> <mi>l</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>J</mi> </munderover> <msub> <mi>ly</mi> <mrow> <mi>j</mi> <mi>l</mi> </mrow> </msub> <mo>,</mo> <mo>&ForAll;</mo> <mi>j</mi> <mo>&Element;</mo> <mi>J</mi> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>13</mn> <mo>)</mo> </mrow> </mrow>
pi represents the reverse order of the processing sequence of the workpiece after the machine serial number is ignored;
the expression of the feasible domain of pi is shown as the formula (14):
the TFT is expressed as the inner product of pi and p, and the expression is shown as the formula (15):
f(π,p)＝f(X,p)＝πTp； (15)
2-2) solving an inner layer problem;
after the decision variable X is converted into pi in the step 2-1), according to E [ f (pi, p)]＝πTE[p]The inner layer problem in the DR-PMSP-MU1 model shown in equation (10) is simplified as shown in equation (16):
wherein, mu-E [ p ] is a real mean vector of all the workpiece processing time, and s.t. represents a constraint condition;
solving equation (16) to obtain the optimal solution ofThe optimal value of the inner layer problem in the corresponding DR-PMSP-MU1 model is
2-3) equivalently converting the DR-PMSP-MU1 model into a solvable model DR-PMSP-MU 2;
replacing the inner layer problem in the formula (10) with the optimal value obtained in the step 2-2), equivalently converting the DR-PMSP-MU1 model into a DR-PMSP-MU2 model, wherein the expression of the DR-PMSP-MU2 model is shown in the formula (17):
<mrow> <munder> <mrow> <mi>m</mi> <mi>i</mi> <mi>n</mi> </mrow> <mrow> <mi>&pi;</mi> <mo>&Element;</mo> <mo>&Pi;</mo> </mrow> </munder> <msup> <mi>&pi;</mi> <mi>T</mi> </msup> <msub> <mi>&mu;</mi> <mi>s</mi> </msub> <mo>+</mo> <mi>&gamma;</mi> <msqrt> <mrow> <msup> <mi>&pi;</mi> <mi>T</mi> </msup> <msub> <mo>&Sigma;</mo> <mi>s</mi> </msub> <mi>&pi;</mi> </mrow> </msqrt> <mo>;</mo> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>17</mn> <mo>)</mo> </mrow> </mrow>
3) solving the DR-PMSP-MU model to obtain an optimal production scheduling scheme;
solving the DR-PMSP-MU2 model shown in the formula (17), wherein the obtained optimal solution of the DR-PMSP-MU2 model is the optimal solution of the DR-PMSP-MU model; the optimal solution is an optimal vector pi value, and the vector pi value is equivalently converted into an optimal two-dimensional matrix Y value containing the processing sequence corresponding to each workpiece through the corresponding relation shown in the formula (13), so that an optimal production scheduling scheme is obtained.
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