CN109918827B - Upper and lower limit convergence search simulation optimization calculation method with screening mechanism - Google Patents

Upper and lower limit convergence search simulation optimization calculation method with screening mechanism Download PDF

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CN109918827B
CN109918827B CN201910207267.0A CN201910207267A CN109918827B CN 109918827 B CN109918827 B CN 109918827B CN 201910207267 A CN201910207267 A CN 201910207267A CN 109918827 B CN109918827 B CN 109918827B
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王嘉宏
吴晓晶
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Fujian University of Technology
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Abstract

The invention relates to an upper and lower limit convergence search simulation optimization calculation method with a screening mechanism. The method comprises the steps of taking the overall service satisfaction degree of a maximized service system as an optimization target, establishing a mixed integer type optimization model of two heterogeneous service type service systems with a screening mechanism, then changing the upper bound or the lower bound of searching in each iteration calculation by utilizing the upper bound and the lower bound to continuously converge the simulation value of the searching system in a systematization mode, and finding out a new gate threshold value through a dichotomy until the convergence reaches a waiting time value which accords with a confidence interval. The method can find out the approximate optimal solution of the optimization model under the limits of large solution space, time and sampling cost.

Description

Simulation optimization calculation method for upper and lower limit convergence search with screening mechanism
Technical Field
The invention relates to an upper and lower limit convergence search simulation optimization calculation method with a screening mechanism.
Background
The simulation optimization method in the prior art focuses on developing a relevant theory how to converge to an optimal solution when the number of samples approaches infinity, and for a simulation optimization model, in the limited number of sample samples, all feasible solutions meeting a random restriction formula cannot be found out.
However, in practice, the size of the samples is usually determined subjectively by the decision maker, and only a limited number of samples can be used. Meanwhile, the larger the number of samples required by system simulation is, the larger the sampling cost is; and in the case of insufficient sample numbers, simulation optimization cannot guarantee that a feasible solution can be found. In addition, in processing an optimization problem with discrete variables, a sub-gradient function (subvariant) needs to be estimated, and research work in the past literature often uses a finite difference method (FiniteDifferences), which often requires a significant computational simulation time cost.
Therefore, the present invention mainly utilizes the optimization skill of the random search algorithm, utilizes the upper and lower bounds to continuously converge the search system simulation value through the systematization mode, changes the upper or lower bounds of the search in each iteration calculation, and finds out a new gate threshold value through the dichotomy until the value converges to the waiting time value which accords with the confidence interval.
Disclosure of Invention
The invention aims to provide a simulation calculation method for convergence search of upper and lower limits with a screening mechanism, which can find out an approximately optimal solution of an optimization model under the limitations of large solution space, time and sampling cost.
In order to achieve the purpose, the technical scheme of the invention is as follows: a convergence search simulation optimization calculation method for upper and lower limits with a screening mechanism is provided, which takes the overall service satisfaction of a maximized service system as an optimization target and establishes a mixed integer type optimization model of two heterogeneous service type service systems with the screening mechanism; and then, calculating the optimal value of the overall service satisfaction degree of the service system by using an upper limit and a lower limit convergence searching optimization algorithm.
In an embodiment of the present invention, the mixed integer optimization model of the two heterogeneous service type service systems with the screening mechanism is as follows:
an objective function: d1·R1(τ)+d2·R2(τ)
The restriction formula 1: w (τ, s)1,s2)≤ε
The restriction formula 2:
Figure GDA0003628706020000011
restriction formula 3: tau is more than 0 and less than 1
Restricted formula 4: s1,s2∈positive integer
Wherein d is1、d2Screening the distribution proportion of the service equipment distributed to the two service types by the customer through the service classification threshold value; r1(τ)、R2(τ) represents satisfaction degrees of the service devices of the two service types, respectively; the limit formula 1 represents the customer expected waiting time W (τ, s)1,s2) Less than a preset expected waiting time threshold epsilon; the left side of the constraint 2 represents the sum of the service personnel of the service equipment, the configuration cost of the service equipment and the operation cost, and the right side represents the total budget B, beta1、β2Respectively representing service equipment using two service types per yearDepreciation amortization cost, p (τ) represents the probability of assignment of a customer to a service device of the second service type, ciI is 1 and 2 respectively represent the cost of hiring service personnel of the two service types; the limiting formula 3 represents the value range of the service classification threshold value tau; restricted formula 4 represents s1、s2The number of service personnel of the service equipment of the two service types is a positive integer.
In an embodiment of the present invention, the specific implementation process of calculating the optimal value of the overall service satisfaction of the service system by using the upper and lower bound convergence search optimization algorithm is as follows:
step S1, according to the restriction formula 2, selecting S1、s2A set C of all possible combinations;
step S2, setting the iteration number n of the optimized solution to 0;
step S21, getting a set of solutions (S) from the set C1 (n),s2 (n)) Setting an algorithm iteration number k as 0;
step S22, when
Figure GDA0003628706020000021
When it is used, order
Figure GDA0003628706020000022
And step S25 is entered, where Δ τ is a predetermined small enough positive value; if not, then
Figure GDA0003628706020000023
Then, set k ← k +1 and
Figure GDA0003628706020000024
proceeding to step S23;
step S23, according to taukPerforming M repeated simulation tests, and recording the obtained waiting time Wj(s1 (n),s2 (n)) J ═ 1,2,3,. M, and the average was calculated:
Figure GDA0003628706020000025
and calculating
Figure GDA0003628706020000026
95% confidence interval of
Figure GDA0003628706020000027
Wherein γ is the half-length of the confidence interval;
step S24, if the preset expected waiting time threshold epsilon satisfies the inequality
Figure GDA0003628706020000028
Or when the number of algorithm iterations K equals K, then set
Figure GDA0003628706020000029
And update the set C ← C \ {(s)1 (n),s2 (n)) }; otherwise, if the desired wait time threshold value is preset
Figure GDA00036287060200000210
At the time, setτ＝τkAnd go back to step S22 to continue execution; if the expected waiting time is preset, the valve value is controlled
Figure GDA0003628706020000031
Then set it
Figure GDA0003628706020000032
And returns to step S22 to continue execution;
step S25, when the set C is equal to Φ, the upper and lower bounds convergence search algorithm loop ends and proceeds to step S3; otherwise, setting an iteration number n ← n +1 of the optimized solution, and returning to the step S21 to continue execution;
step S3: solutions obtained for upper and lower bound convergence search algorithms
Figure GDA0003628706020000033
Determining the feasibility of the solution by using a feasibility verification program to delete the infeasible solution; let C*Represents a passing throughAll selected by the verification program
Figure GDA0003628706020000034
A solution set of compositions; finally outputting the optimal expected waiting time gate valve value
Figure GDA0003628706020000035
And the optimum value SL (τ) that can calculate the overall service satisfaction of the service system*)。
Compared with the prior art, the invention has the following beneficial effects:
1. the simulation optimization algorithm provided by the invention analyzes the complexity optimization problem existing in a random system in a mode of constructing a prediction model, and finds out a system with the best expected performance from a plurality of different systems through a simulation experiment mode on the basis of a statistical theory so as to provide a decision maker for reference.
2. The simulation optimization algorithm provided by the invention can analyze the complex problem which can not be analyzed by a general mathematical model, and can analyze important factors and confidence level which influence the system performance through experimental design.
3. The invention develops an efficient simulation optimization algorithm to solve an optimization problem with randomness, and the simulation optimization algorithm provided by the invention can be written into a software program, and scientific calculation and system simulation are carried out through a computer to efficiently solve the random optimization problem.
Drawings
FIG. 1 is a diagram of two heterogeneous service type service systems with a screening mechanism according to the present invention.
Detailed Description
The technical scheme of the invention is specifically explained below with reference to the accompanying drawings.
The invention provides a simulation optimization calculation method for upper and lower limit convergence searching with a screening mechanism, which takes the overall service satisfaction degree of a maximized service system as an optimization target and establishes a mixed integer optimization model of two heterogeneous service type service systems with the screening mechanism; and then, calculating the optimal value of the overall service satisfaction degree of the service system by using an upper limit and a lower limit convergence searching optimization algorithm.
Specifically, the present invention has a screening mechanism, which can distinguish two service devices with different service types according to the threshold value of the service classification gate, as shown in fig. 1. It is desirable to maximize service satisfaction of the entire service system by adjusting this threshold value while satisfying the latency constraint. Through the information system, the value is assigned in advance according to the type characteristics of the customers and then is distributed to the service equipment of different service types, so that the safety level of the service system can be maximized under the condition of meeting the allowable waiting time and the total budget limit. If the customer can be properly assigned to the service equipment with the proper service type, the service satisfaction degree of the service system can be maximized, and the service efficiency requirement can be met.
In the present invention, a mixed integer optimization model with random restriction is solved as follows:
maximizing the objective function: d1·R1(τ)+d2·R2(τ)
Satisfies the following conditions:
the restriction formula 1: w (τ, s)1,s2)≤ε
The restriction formula 2:
Figure GDA0003628706020000041
restriction formula 3: tau is more than 0 and less than 1
Restricted formula 4: s1,s2∈positive integer
Wherein d is1、d2Screening the distribution proportion of the service equipment distributed to the two service types by the customer through the service classification threshold value; r1(τ)、R2(τ) represents satisfaction degrees of the service devices of the two service types, respectively; the limit formula 1 represents the customer expected waiting time W (τ, s)1,s2) Less than a preset expected waiting time threshold epsilon; the left side of the constraint 2 represents the sum of the service personnel of the service equipment, the configuration cost of the service equipment and the operation cost, and the right side represents the total budget B, beta1、β2Service device respectively representing annual use of two service typesDepreciated amortization cost, p (τ) represents the probability of assignment of a customer to a service device of a second service type, ciI is 1 and 2 respectively represent the cost of hiring service personnel of the two service types; the limiting formula 3 represents the value range of the service classification threshold value tau; restricted formula 4 represents s1、s2The number of service personnel of the service equipment of the two service types is a positive integer.
The modeling concept of this mathematical model is described as follows:
1. the objective function of the optimization model is to maximize the overall service satisfaction of the service system: obtaining customer distribution ratio d by screening expected waiting time threshold value tau1、d2Weighted summation of satisfaction R of two different service types1(τ)、R2(τ)。
2. The first constraint guarantees the service efficiency requirement: expected wait time W (τ, s) for customer1,s2) Needs to be less than a desired latency threshold epsilon.
3. The second constraint is the total budget constraint: the sum of the service personnel of the service equipment, the configuration costs and the operating costs of the service equipment cannot exceed a given total budget B.
4. The third constraint specification expects an assigned range for the latency gate threshold τ: the adjustment of the desired latency gate threshold τ needs to be between 0% and 100%.
5. The fourth constraint expression indicates that the number of service personnel and the number of service equipment configured for the two service types of service equipment are positive integers.
6. The mathematical optimization model takes the overall service satisfaction degree of a maximum service system as an optimization target, one of decision variables is an expected waiting time threshold value tau of customers to be classified, which are assigned to different service types of service equipment, and the decision variables belong to continuous variables; and the other decision variable is the configuration number s of service personnel and service equipment of two service types1、s2Belong to discrete variables.
The first attached table and the second attached table summarize the related symbol definitions of a mixed integer optimization model with random restriction to be processed by the technology of the invention.
Auxiliary table I, decision variable and performance index summary table of optimization model
Figure GDA0003628706020000051
Parameter symbol summary table of attached table two and optimization model
Parameter(s) Definition of
d1 The service efficiency of the service equipment of service type I is a constant between 0 and 1.
d2 The service efficiency of the service equipment of service type II is a constant between 0 and 1.
c1 Service type I the cost of hiring a service person.
c2 Type II service hiring a service person.
ε The customer expects an efficiency requirement value for the wait time.
B For building service equipment and hiring service personnelThe total budget amount.
β1 Using service equipment of service type I annually depreciates costs.
β2 Using service type II service equipment annually depreciates costs.
p(τ) When the classification threshold is τ, the customer is assigned an allocation probability of service type II.
R1(τ) The customer's service satisfaction at service type I when the classification threshold is τ.
R2(τ) And when the classification threshold is tau, the service satisfaction degree of the customer in the service type II is realized.
In this mixed integer optimization model with random restriction, the mathematical function R1(τ)、R2(τ), p (τ) and W (τ, s)1,s2) When the system can not be expressed in a mathematical analytic expression, the invention provides a simulation optimization algorithm, finds out a mathematical function estimated value through a system simulation method, and calculates the service classification door threshold value tau and the personnel configuration numbers and s of two security check stations1、s2The optimal solution of (1).
Even sometimes in some special cases, the function R1(τ)、R2(τ), p (τ) and W (τ, s)1,s2) The function value can be calculated by a mathematical analysis mode, but the function is too complex and difficult to calculate by a computer, so that the optimization model is obtainedThe solution time increases substantially. The problem of random optimization is that a function value cannot be expressed by a mathematical expression, and the problem can only be solved by using a simulation optimization algorithm provided by the invention.
Specifically, the invention is realized by the following modes:
the upper and lower bound convergence search optimization algorithm provided by the invention utilizes the simulation value of the upper and lower bound continuous convergence search waiting time, changes the upper or lower bound in each iteration, and finds out a new service classification gate threshold value through a dichotomy until the upper and lower bound continuous convergence search waiting time value meets the confidence interval. The upper and lower bound convergence search algorithm continues to perform the following main procedure steps until the best service classification threshold τ is output*. The parameter notation of the algorithm is as follows: let K be the number of algorithm iterations, K be the maximum number of iterations,
Figure GDA0003628706020000061
to serve the upper bound of the classification gate threshold,τfor the lower bound of the service classification threshold, M is the number of repetitions of the simulation test, θ1、θ2The gradient parameters of the solution are refined for each iteration.
Main program of upper and lower bound convergence search algorithm:
step S1, according to the restriction formula 2, selecting S1、s2A set C of all possible combinations;
step S2, setting the iteration number n of the optimized solution to 0;
step S21, getting a set of solutions (S) from the set C1 (n),s2 (n)) Setting an algorithm iteration number k as 0;
step S22, when
Figure GDA0003628706020000062
When it is used, order
Figure GDA0003628706020000063
And proceeds to step S25, where Δ τ is a predetermined sufficiently small positive value; if not, then
Figure GDA0003628706020000064
Then, set k ← k +1 and
Figure GDA0003628706020000065
proceeding to step S23;
step S23, according to taukPerforming M repeated simulation tests, and recording the obtained waiting time Wj(s1 (n),s2 (n)) J ═ 1,2,3,. M, and the average was calculated:
Figure GDA0003628706020000066
and calculating
Figure GDA0003628706020000067
95% confidence interval of
Figure GDA0003628706020000068
Wherein γ is the half-length of the confidence interval;
step S24, if the threshold value epsilon of the preset expected waiting time door satisfies the inequality
Figure GDA0003628706020000069
Or when the number of algorithm iterations K equals K, then set
Figure GDA00036287060200000610
And update the set C ← C \ {(s)1 (n),s2 (n)) }; otherwise, if the desired wait time threshold value is preset
Figure GDA0003628706020000071
At the time, setτ＝τkAnd go back to step S22 to continue execution; if the expected waiting time is preset, the valve value is controlled
Figure GDA0003628706020000072
Then set it
Figure GDA0003628706020000073
And returns to step S22 to continue execution;
step S25, when the set C is equal to Φ, the upper and lower bounds convergence search algorithm loop ends and proceeds to step S3; otherwise, setting an iteration number n ← n +1 of the optimized solution, and returning to the step S21 to continue execution;
step S3: solutions obtained for upper and lower bound convergence search algorithms
Figure GDA0003628706020000074
Judging the feasibility of the solution by using a feasibility verification program to delete an infeasible solution; let C*All selected by the feasibility verification program are shown
Figure GDA0003628706020000075
A solution set of compositions;
and (3) outputting an algorithm: the optimal expected waiting time gate valve value can be obtained through the upper and lower bound convergence searching algorithm
Figure GDA0003628706020000076
And the optimum value SL (τ) that can calculate the overall service satisfaction of the service system*)。
Reference documents:
【1】 Reference to Ranking and Selection program (Ranking and Selection): andrad' otter, S., Kim, S.H, (2010) full sequential processes for compressing constrained systems via simulation logic, vol.57, pp.403-421.
The above are preferred embodiments of the present invention, and all changes made according to the technical scheme of the present invention that produce functional effects do not exceed the scope of the technical scheme of the present invention belong to the protection scope of the present invention.

Claims (1)

1. A convergence search simulation optimization calculation method of upper and lower limits with a screening mechanism is characterized in that a mixed integer type optimization model of two heterogeneous service type service systems with the screening mechanism is established by taking the overall service satisfaction of a maximized service system as an optimization target; then, calculating the optimal value of the overall service satisfaction of the service system by using an upper and lower limit convergence searching optimization algorithm;
the mixed integer type optimization model of the two heterogeneous service type service systems with the screening mechanism is as follows:
an objective function: d is a radical of1·R1(τ)+d2·R2(τ)
The restriction formula 1: w (τ, s)1,s2)≤ε
The restriction formula 2:
Figure FDA0003628706010000011
restriction formula 3: tau is more than 0 and less than 1
Restricted formula 4: s is1,s2∈positive integer
Wherein d is1、d2Screening the distribution proportion of the service equipment distributed to the two service types by the customer through the service classification threshold value; r1(τ)、R2(τ) represents satisfaction degrees of the service devices of the two service types, respectively; the limit formula 1 represents the customer expected waiting time W (τ, s)1,s2) Less than a preset expected waiting time threshold epsilon; the left side of the restriction 2 represents the sum of the service personnel of the service equipment, the configuration cost of the service equipment and the operation cost, and the right side represents the total budget B, beta1、β2Respectively representing depreciation costs for using service devices of the two service types per year, p (τ) representing the distribution probability of a customer being assigned to a service device of the second service type, ciI is 1 and 2 respectively represent the cost of hiring service personnel of the two service types; the limiting formula 3 represents the value range of the service classification threshold value tau; restricted formula 4 represents s1、s2The service personnel number of the service equipment of the two service types is a positive integer;
the specific implementation process of calculating the optimal value of the overall service satisfaction of the service system by using the upper and lower limit convergence searching optimization algorithm is as follows:
step S1, according to the restriction formula 2, selecting S1、s2A set of all combinations C;
step S2, setting the iteration number n of the optimized solution to 0;
step S21, getting a set of solutions (S) from the set C1 (n),s2 (n)) Setting an algorithm iteration number k as 0;
step S22, when
Figure FDA0003628706010000012
When it is used, order
Figure FDA0003628706010000013
And step S25 is entered, where Δ τ is a predetermined positive value; if not, then
Figure FDA0003628706010000014
Then, set k ← k +1 and
Figure FDA0003628706010000015
the process proceeds to step S23, where,
Figure FDA0003628706010000016
to serve the upper bound of the classification gate threshold,τclassifying a lower bound of the threshold value for the service;
step S23, according to taukPerforming M repeated simulation tests, and recording the obtained waiting time Wj(s1 (n),s2 (n)) J ═ 1,2,3,. M, and the average was calculated:
Figure FDA0003628706010000021
and calculating
Figure FDA0003628706010000022
95% confidence interval of
Figure FDA0003628706010000023
Wherein the content of the first and second substances,γ is half the length of the confidence interval;
step S24, if the preset expected waiting time threshold epsilon satisfies the inequality
Figure FDA0003628706010000024
Or when the number of algorithm iterations K equals K, then set
Figure FDA0003628706010000025
And update the set C ← C \ {(s)1 (n),s2 (n)) K is the maximum iteration number; on the contrary, if the expected waiting time threshold value epsilon is preset
Figure FDA0003628706010000026
Time, set upτ＝τkAnd go back to step S22 to continue execution; if the expected waiting time is preset, the valve value is controlled
Figure FDA0003628706010000027
Figure FDA0003628706010000028
Then set it
Figure FDA0003628706010000029
And returns to step S22 to continue execution;
step S25, when the set C is equal to Φ, the upper and lower bounds convergence search algorithm loop ends and proceeds to step S3; otherwise, setting an iteration number n ← n +1 of the optimized solution, and returning to the step S21 to continue execution;
step S3: solution obtained for upper and lower bound convergence search algorithm
Figure FDA00036287060100000210
Judging the feasibility of the solution by using a feasibility verification program to delete an infeasible solution; let C*All selected by the feasibility verification program are shown
Figure FDA00036287060100000211
A solution set of compositions; finally outputting the optimal expected waiting time gate valve value
Figure FDA00036287060100000212
And calculates the optimum value of the overall service satisfaction of the service system as SL (tau)*)。
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