CN109685244A - The continuous site selecting method of reliability of return is considered under limited information scene - Google Patents
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Abstract
The present invention relates to the continuous site selecting methods of reliability that return is considered under a kind of limited information scene, the following steps are included: step 1, site selection model are assumed, step 2, defined variable, step 3, site selection model building, 3.1, the site selection model expression formula in homogeneity continuous level under solution unit area；3.2, the site selection model expression formula in the heterogeneous continuous level S of two dimension is solved according to 3.1 result, finally obtains the optimal total cost of site selection model and optimal Facilities Construction quantity in the heterogeneous continuous level of two dimension；3.3, discretization is carried out to the optimal total cost of site selection model and optimal Facilities Construction quantity, obtains the optimum results of site selection model.
Description
Technical field
The present invention relates to the optimization layouts of infrastructure, node addressing in Communication and Transportation Engineering, Logistics Engineering, specifically relate to
And the continuous site selecting method of reliability of return is considered under a kind of limited information scene.
Background technique
Either in commercial transport system (such as logistics network) or in public transportation system, how to select optimal
Facility locations are all the matter of utmost importance and central decision problem of entire planning system.The essence of location problem is exactly specified
Region in build certain amount facility, to meet the service request of all customers in the region, in this process, meeting
The one-time construction expense of facility and the long-term freight of customer are generated, as shown in Figure 1: so-called Optimizing Site Selection model, just
It is that service facility is built in most economical mode, to obtain relatively optimal system performance.It is well known that being built in site selection model
Being set as this, there is benefits to carry on the back adverse effect between long-term transportation cost, in simple terms, if we are intended to reduce construction cost,
The quantity for so building facility is necessarily reduced, and the average distance of customer to facility increases, and eventually leads to transportation cost increase；Instead
It, if we will reduce transportation cost, it is meant that customer's arrival service facility to be more easier must just be built more
Facility, so as to cause higher construction cost.This relationship is quite similar to balance, if a side is reduced, being equivalent to centainly be will lead to
The increase of another party, and the purpose optimized is that and finds an equalization point, so that sum of the two is minimum.
It is existing the study found that facility may all be damaged either in natural calamity or human accident, meaning
The factor considered in addressing is known other than the operating of the high efficiency of system, it is also contemplated that the reliability of system.It solves
The classical way of reliability location problem is the thinking using stand-by facility, i.e., was proposed by Snyder and Daskin in 2005
The reliability facility site selection model of constant expense.
The prior art one: existing reliability site selection model have a basic assumption, also referred to as Complete Information it is assumed that
That is, each customer has a main services facility and several active service facilities, fail when main services facility
When situation, which immediately knows that the real time status information of facility, by accessing other available active service facilities to obtain
Service.Customer will obtain the real time status information of facility, so as to according to the priority level distributed, from the location of oneself
The position for directly reaching an effective facility, as shown in Fig. 2 (a).But in reality, since technical barrier, information propagate failure
Etc. reasons, this hypothesis be not set up always, therefore the prior art two have also been proposed be more in line with reality incomplete letter
Breath is assumed, that is, customer can not learn the information of facility real-time running state, he will access one by one according to prespecified order
Neighbouring facility is in the facility of normal operation state and is serviced, or accessed all specified facilities until finding
It has been found that whole damages, then abandon servicing and therefore receives punishment cost, as shown in Fig. 2 (b).
But it is had the disadvantage in that in the prior art two
(1) the passenger traffic phenomenon that the prior art is directed in traffic is inquired into, and what is obtained is comprehensive passenger transport hub
Optimization Method for Location-Selection does not discuss for the shipping phenomenon in traffic, has ignored the return cost behavior of cargo transport.Cause
For the particularity of movement operation, when the activity end --- after cargo joins, delivery vehicle will generally return to Original Departure Point,
That is, the transportation cost of return is existing.
(2) model that the prior art two constructs is a kind of discrete model, it is longer to acquire the time accurately solved, therefore can only answer
For small-scale or medium-scale scene, to large scale scene, big data processing capacity is poor, can not ask within the regular hour
It obtains and accurately solves.
Summary of the invention
In view of the deficiencies in the prior art, the purpose of the present invention is to provide consider under a kind of limited information scene
The continuous site selecting method of the reliability of return constructs one kind under imperfect information scene, increases the round-trip double of transport return cost
The transit node reliability Continuous Location of journey path mode, that is, CRLP-IITT model.
To achieve the above objectives, the technical solution adopted by the present invention is that:
The continuous site selecting method of reliability of return is considered under a kind of limited information scene, comprising the following steps: step 1,
Site selection model is assumed
It (1) is not that permanently can be used, there are certain probability of damage after transit node construction is come into operation；
(2) probability of damage is independent from each other between each transit node；
(3) client only knows the initial information of transit node, in this case it is not apparent that the real time information of transit node；
(4) client will successively access prevailing traffic node and spare transit node according to assignment order, and can only access
The transit node of client oneself is distributed to, regardless of whether being serviced, will not all visit again other transit nodes；
(5) if client finally there is no service, then client can only abandon and receive certain rejection penalty；
(6) no matter whether client is serviced, and will finally return to Original Departure Point, that is, there are return costs；
Step 2, defined variable
In transportation service region, the heterogeneous continuous level of two dimension is set as S, and arbitrary point position is set as x, the visitor at the x of position
Family transportation service demand is usedIt indicates；Transit node builds the arbitrary point position x in the transportation service region
Place, fixation construction cost are f (x), and the probability of damage for the transit node built at the x of position is q (x), and client is in position x
Abandon the punishment cost that transportation service obtains in placeThe grade that client accesses certain transit node is r, and each client accesses fortune
The maximum access number of defeated node is defined as R, with client LxIndicate the client on the x of position, transit node ZxIt indicates in position x
On node；
Step 3, site selection model building
3.1, the site selection model expression formula under unit area is solved in homogeneity continuous level；
Homogeneity continuous level is a unlimited two-dimensional space, the related ginseng in homogeneity continuous level, on all positions
Numerical value is identical and fixation, thenWherein f indicates transport
The constant of node construction cost, λ indicate the constant of client's transportation service demand,Indicate that the constant of punishment cost, q indicate transport
The constant of node probability of damage；
(1) analyzed area is determined
Assuming that the initial coverage of each transit node is the mesh space being made of regular hexagon, area A, and
And each facility is all located at the center of regular hexagon, and regular hexagon is divided into 12 equal small triangles, chooses wherein one
A small triangle is as analyzed areaAnalyzed areaArea be
(2) the site selection model expression formula under unit area is determined
1. solving the Facilities Construction cost C under unit areaF；
2. solving the punishment cost C under unit areaP；
3. solving the transportation cost C under unit areaT；
To the Facilities Construction cost C under unit areaF, punishment cost C under unit areaPWith the transport under unit area
Cost CTSummation, obtains the site selection model expression formula under unit area；
3.2, the site selection model expression formula in the heterogeneous continuous level S of two dimension is solved according to 3.1 result, finally obtained
The optimal total cost of site selection model and optimal Facilities Construction quantity in the heterogeneous continuous level of two dimension；
3.3, discretization is carried out to the optimal total cost of site selection model and optimal Facilities Construction quantity, obtains site selection model
Optimum results.
Facilities Construction cost C on the basis of above scheme, under unit areaFCalculation formula it is as follows:
CF=f/A (10).
Punishment cost C on the basis of above scheme, under unit areaPCalculation formula it is as follows:
In formula, qR+1Represent the probability that client in unit area does not obtain service.
Transportation cost C on the basis of above scheme, under unit areaTWith feasible solution CT-FSIt indicates, calculation formula is as follows
It is shown:
In formula, ω is freight rates, wherein βrFor from the facility of r grade to analyzed areaDistance proportion coefficient, β0
For from the facility of 0 grade to analyzed areaDistance proportion coefficient, βRFor from the facility of R grade to analyzed areaDistance
Proportionality coefficient.
On the basis of above scheme, the site selection model expression formula under unit area is as follows:
In formula, C (A) represents the total cost of unit area in homogeneity continuous level.
The parameters f (x), λ in the heterogeneous continuous level S of two dimension are assumed on the basis of above scheme, in step 3.2
(x),Relatively slow with the variation of q (x), the area of initial service area of the Facilities Construction at the x of position is with continuously
FunctionIt indicates, then the expression formula of site selection model is as follows in two-dimentional heterogeneous continuous level:
In formula, C (x, A (x)) represents the total cost of unit area in the heterogeneous continuous level of two dimension；
To formula (26) derivation, service area function A optimal at x is obtained*(x), it is then obtained using the method for integral
The optimal total cost C of the heterogeneous continuous level S of two dimension*With optimal Facilities Construction quantity N*, calculation formula is as follows:
C*=∫x∈SC(x,A*(x))dx (27)
N*≈∫x∈S[A*(x)]-1dx (28)。
On the basis of above scheme, the process of discretization algorithm is as follows:
3.3.1, planning region PS, PS ∈ S are initialized as chimb circle first；
3.3.2, the region division for surrounding chimb circle is multiple cellulars, and the size of cellular is w × w, and the center of cellular is sat
Mark (xi,yi) indicate, wherein i=1,2,3 ..., I indicate that the number of cellular calculates the service of its client for any cellular i
Radius g (xi,yi), formula is as follows:
A in formula*(xi,yi) indicate in (xi,yi) at optimal service area function；
3.3.3, NC cellular is randomly selected, the center of cellular is enabled to indicate the position of node addressing, using nc=1,
2,3 ..., NC, nj=1,2,3 ..., NC indicate the number of node, i.e. cellular nc, nj indicates the cellular comprising node, enable iteration time
Number m=1；
3.3.4, the repulsive force between each cellular comprising node and the cellular comprising node are calculated and between boundary
Repulsive force；
Cellular nc comprising node is h by the repulsive force of the cellular nj comprising nodeRF(nc, nj), calculation formula is as follows:
Wherein h indicates maximum repulsive force, lnc-njIt indicates between the cellular nc comprising node and the cellular nj comprising node
Distance, formula are as follows:
hRFThe component of (nc, nj) in x-axis and y-axis direction is respectively as follows:
Cellular nc comprising node and the repulsive force between boundary are hB, calculation formula is as follows:
hB=(NC+1) h (34)
hBComponent in x-axis and y-axis direction is respectively as follows:
All component to the cellular comprising node in x-axis and y-axis direction are summed, and are obtained in x-axis and y-axis direction institute
The resultant force size received, uses H respectivelyx(nc) and Hy(nc) it indicates, if all cellular stress comprising node are 0, goes to
Step 3.3.7；Otherwise step 3.3.5 is carried out；
3.3.5, according to the resultant force size suffered by x-axis and y-axis direction of the cellular comprising node, the member comprising node is calculated
It is found out and suffered conjunction according to 8 directions in the direction of suffered resultant force and the cellular surrounding comprising node in the direction of resultant force suffered by born of the same parents
Target of the cellular as the cellular movement comprising node on closest direction is chosen in the closest direction in the direction of power
Cellular；
3.3.6, the cellular comprising node that all stress are not zero is moved in corresponding target cellular, m=m
+1；If m < mmax, re-start step 3.3.4；Otherwise, step 3.3.3 is re-started；
3.3.7, according to the obtained cellular position comprising node, each node is obtained using Thiessen polygon method
Service area boundary.
Detailed description of the invention
The present invention has following attached drawing:
Fig. 1 location problem description figure.
Fig. 2 is completely with the access route map under imperfect information, and wherein Fig. 2 (a) is that Complete Information accesses route, Fig. 2 (b)
Route is accessed for imperfect information.
Fig. 3 client accesses order schematic diagram.
Fig. 4 analyzed area schematic diagram.
Fig. 5 analyzed area and facility number schematic diagram.
The error schematic diagram of Fig. 6 feasible solution and lower bound.
Fig. 7 planning region PS and its boundary schematic diagram.
Fig. 8 target cellular chooses schematic diagram.
The flow chart of Fig. 9 discretization algorithm.
Figure 10 Beijing Tianjin and Hebei Region and discrete addressing point distribution schematic diagram.
Figure 11 continuous level planning region schematic diagram.
Customer demand Density Distribution schematic diagram in Figure 12 continuous level.
Figure 13 continuous level interior nodes construction cost distribution schematic diagram.
Figure 14 continuous level interior nodes probability of damage distribution schematic diagram.
Specific embodiment
Below in conjunction with attached drawing, invention is further described in detail.
1, model hypothesis
By the description to region transportation network location problem under information failure situation, referring to the vacation of traditional site selection model
If the basis of CRLP-IITT model is assumed to be described as follows by the present invention:
It (1) is not permanent available, but there are certain failure probabilities after node construction is come into operation；
(2) probability of damage is independent from each other between each node；
(3) client only knows the initial information (node location, capacity etc.) of node, in this case it is not apparent that the real time information of node,
That is, just can know that whether node is in effective status when client only arrives at certain node on the spot, can oneself be serviced；
(4) client will successively access main node and standby node according to assignment order, and can only access and distribute to him
Node will not all visit again the node except distribution regardless of whether serviced, that is, the node number of client's access is that have
Limit；
(5) if client finally there is no service (all nodes for distributing to the client all fail), then client is only
It can abandon and receive certain rejection penalty；
(6) no matter whether client is serviced, and will finally return to Original Departure Point, that is, there are return costs；
2, defined variable
According to assumed above, the description of foundation CRLP-IITT problem, it is convenient for modeling to simplify problem, the present invention should
Problem is described below from mathematical angle: in coverage, the heterogeneous continuous level of two dimension is set as S, and arbitrary point position is set as x,
Customer service requirement at the position is usedIt indicates；Node can build any position x in the planning region
Place, fixation construction cost are f (x), and the node probability of damage of place construction is q (x), and client abandons service here and obtains
Punishment cost bePresent invention assumes that the case where being damaged between different nodes is independent from each other；Client can not be real
When grasp node state (work whether), therefore they need using " trial and error strategy " successively accessed node, to determine whether to obtain
It must service, the access order in this model is also one of decision variable, and the grade that client accesses certain node is r, and each client is most
Big accessed node number is set to R.
With client LxIndicate the client on the x of position, transit node ZxIndicate the node on the x of position.
3, model construction
In traditional site selection model, totle drilling cost generally comprises the one-time construction expense of node and the traffic expense that client is long-term
With the present invention follows the thinking.Simultaneously, it is contemplated that the integrality and mathematical expression identity of model construction, the present invention will also punish
Penalize cost that totle drilling cost is added, that is, when client can not finally be serviced, it has to when abandoning, will receive certain rejection penalty.
This point is also to be consistent with reality, it is however generally that, cargo has certain delivery date and shelf-life, if client can not obtain in time
It must service, cargo cannot be sent in junior's consumer's hand, additional loss can be generated, this partial loss is in this model with punishing
Expense is penalized to indicate.
Decision variable of the invention includes addressing point position and the access order of client of node, when two variables are deposited simultaneously
When, to case study, there are biggish difficulty, and therefore, according to the decomposition thinking of existing document, the present invention assumes initially that addressing
Point it has been determined that continue the access order optimization problem of discussion client on this basis.
With x={ x0,x1,x2,…,xNIndicate the facility locations set built, (N+1) indicates facility sum.Thus
The construction cost of available facility
In operation stage, all facility moment built all suffer from uncertain event and occur, and in turn result in and set
The damage applied cannot be run.Once facility is damaged, client cannot timely obtain the status information of facility, therefore client needs
It to be serviced using " trial and error " strategy, i.e., client successively accesses all facilities for distributing to him according to set order until visiting
Ask the facility or abandon seeking to service that first is working.
J (x | x) it indicates to distribute to client L in the facility collection x builtxThe collection with different priority levels facility
It closes, concentrates element in the facilityIndicate client LxIn the facility that r-th of grade is assigned to.Such as
The fruit facility works normally, lower than the facility cisco unity malfunction of the grade.As client LxReach facility jr(x) it and is serviced
When, the order that he successively accesses facility is as shown in Figure 3.
In this process, client LxFrom starting point, facility j is successively accessed0(x),j1(x) ..., until he is from facility
jr(x) it is serviced.Client L during being somebody's turn to doxTravel distance be known as backhaul, transport costIt indicates；Later, objective
Family LxHis starting point, return transport cost can directly be returnedIt indicates.
Use dr(J (x | x)) indicate client LxFrom starting point to the transport cost for returning to starting point, the expense is by going
Journey transport costWith backhaul transport costIt is formed.The traffic expense of unit distance is indicated with ω
With also referred to as freight rates.
The present invention indicates the travel distance between any two points using Euclidean distance formula, it is hereby achieved that dr(J(x|
X)),AndExpression formula
In formula,Indicate client LxAccessed facility j0(x) position；Indicate client LxAccessed facility jt-1
(x) position；Indicate client LxAccessed facility jt(x) position；Indicate client LxAccessed facility jr(x) position
It sets；
In view of all possible facility damages scene, using Pr(J (x | x)) indicate to work as facility jr(x) in running order
When other are in damaged condition lower than all facilities of grade r, client LxFrom facility jr(x) probability of service is obtained.It indicates when the damage of all facilities, client LxThe probability of any service is not obtained.For
Client LxFor, his expectation transport cost is
Wherein, dR(J (x | x)) indicate transport cost as r=R, the first part in formula (5) indicates that client Lx is obtained
Expectation transport cost when must service, second part indicate client LxTransport cost when service is not obtained finally.Using integral
Method, can get the total expectation transport cost of all clients in planning region, it is as follows:
As client LxWhen finally not obtaining service, he needs additionally to bear certain rejection penalty.UsingIndicate visitor
Family LxUnit demand rejection penalty, then his rejection penalty can be expressed as
The same method using integral, can obtain the total expectation punishment cost of all clients in planning region, following institute
Show:
According to the analysis of above-mentioned cost structure, the mesh of the lower reliability Continuous Location for considering return expense of information failure
Mark is by choosing suitable construction facility set x and corresponding access facility set { J (x | x) }x∈STo obtain the smallest system
Total cost obtains the totle drilling cost expression formula of continuous model are as follows:
It is the analytic process of totle drilling cost expression formula above, formula (9) is the CRLP-IITT model of most original, but existing
Model form can not direct solution, therefore deformation simplification is carried out to model, and using continuous approximation method in the hope of close
Like optimal solution.
4, model conversion
Continuous model in location problem is usually to be unable to direct solution.Therefore the present invention will be utilized from mathematical angle
Continuous approximation method converts the formula, obtains the formula that can acquire approximate optimal solution.
The solution throughway of continuous approximation method are as follows: in homogeneity continuous level, the parameters of model be all it is identical, this
When continuous model be easy to get approximate optimal solution；And in limited heterogeneous continuous level, arbitrary point and its surrounding are very small
In region, the opposite variation of parameter is very slow, and total cost is influenced very little by other regions, therefore the region can be approximate
Regard homogeneity continuous level as, and then obtain the region approximate optimal solution；Finally to all areas in heterogeneous continuous level into
Row summation, has just obtained the approximate optimal solution of heterogeneous continuous level Facility Location Problem.
In present invention assumes that, client can only access the node of fixed quantity, and corresponding punishment is born if not obtaining service
Expense, but other nodes will not be accessed, that is, client can only access the node near it, will not access more remote section
Point.This also complies with the basic ideas of continuous approximation method, illustrates to select under information failure scenario in the heterogeneous plane of the invention constructed
Location model is suitble to be converted using continuous approximation method, so as to find out approximate optimal solution.
4.1, homogeneity continuous level location problem
According to continuous approximation method, the present invention considers the location problem in homogeneity continuous level first.
It is known that homogeneity continuous level is a unlimited two-dimensional space, such as:In the plane, on all positions
Related parameter values are identical and fixation, such as:
(1) analyzed area is determined
According to existing research, in continuous approximation method, it is desirable that required node location set and visit in site selection model
It asks order, should determine the initial service coverage area of node first, that is, assuming that individual facilities are taken when all facilities are all effective
The region of business；After acquiring initial service area, node is can be obtained by mathematical methods such as integrals in the density of available node
The access order of position and client.
According to existing research, initial service area be regular hexagon timeliness rate highest, therefore present invention assumes that each node it is initial
Service area constitutes a mesh space being made of regular hexagon, area A, and each facility is all located at positive six side
The center of shape.Due to all regular hexagons be it is identical, choose one of regular hexagon as survey region.
It makes discovery from observation, if regular hexagon is divided into 12 small triangles, as shown in figure 4, due to regular hexagon
Symmetry all triangles are the same.Therefore one of triangle is chosen as final analyzed area
Its area is
(2) unit area totle drilling cost is determined
Due to being homogeneity continuous level, to optimization object function, it is equivalent to optimization analyzed area at this timeInterior list
Plane accumulates totle drilling cost, i.e. unit area system total cost C (A) is minimum.C (A) includes the Facilities Construction cost C of unit areaF, single
The transportation cost C of plane productTAnd the punishment cost C of unit areaP.The expression formula of this three parts cost is discussed separately below.
1. construction cost CF
Analyzed areaIn initial service area in a facility, therefore can easily it obtain in the region
The Facilities Construction expense of unit area, as follows
CF=f/A (10)
2. punishment cost CP
For the rejection penalty of unit of account area, first have to determine that the client in unit area does not obtain the general of service
Rate usesIt indicates.For single client, no matter which kind of access order he must access (R+ when not obtaining service finally
1) a facility.Therefore, regionEach interior client has the identical probability for not obtaining service.Client in unit area
The probability for not obtaining service is equivalent to any clientThe probability of service is not obtained, formula is as follows,
The rejection penalty of unit area may further be obtained, it is as follows
3. transportation cost CT
For the freight of unit area, calculation formula is extremely complex.Under information failure conditions, client is adopted
Trial and error strategy is taken to be serviced, i.e., client successively accesses allocated facility according to set sequence, until being accessed first
Until the facility of normal work is serviced or is abandoned to be serviced.Later, client returns to the departure place of oneself.
Therefore, the freight C of unit areaTTwo parts, respectively the backhaul expense of unit area can be divided intoWith
Return expenseAccording to formula (3) and (4), the backhaul expense of available unit areaWith return expenseExpression
Formula, as follows
In formulaIndicate the backhaul transport cost as r=R,Indicate the backhaul as r=R
Transport cost；
Then, freight is calculated according to formula (13) and (14).
4. totle drilling cost C (A)
C (A)=CF+CT+CP
(3) model solution
It should be noted that client LxFacility distribution set J (x | x) and decision variable, it determines freight
Size.Optimal facility allocation set J (x | x) and access order are only obtained, could can obtain optimal unit area
Freight.
However obtaining optimal facility access order (OVS) is a huge challenge, because of each clientThere is it
Respective optimal facility access order, therefore feasible access order (FVS) is constructed first, approximately substitute optimal access time
Sequence.
1. feasible solution
According to the cost formula of upper trifle, any client L can be obtainedxProbability P (the J of service is obtained from r grade facility
(x | x)), formula is
Firstly, by all facilities in plane since the facility where survey region according to sequence counterclockwise successively into
Row number, as shown in Figure 5.
Assuming that analyzed areaInterior all clients access facility all in accordance with the number order of facility, i.e.,
Thus, it is possible to calculate client L according to the geometric properties of figurexTravel distance, and then obtain transport cost.It is returning
In journey freight, any facility to analyzed areaThe average distance of interior all clients can be obtained using the method for integral
?.As can be known from the results, average distance and A1/2Direct proportionality.Using βrA1/2Indicate the facility of r grade to analyzed areaThe average distance of all clients, wherein βrFor from the facility of r grade to analyzed areaDistance proportion coefficient, β0For from 0
The facility of grade is to analyzed areaDistance proportion coefficient, βRFor from the facility of R grade to analyzed areaDistance proportion
Coefficient.
Therefore, return cost formula is as follows
In backhaul freight, regionThe average distance of interior all clients to its 0th grade facility is β0A1/2。
As grade r > 0, according to the geometric properties of figure, regionInterior all clients are from its (r-1) grade facility to its r etc.
Grade facility average distance be
Thus the formula of backhaul freight is obtained, as follows
This makes it possible to obtain the freight C of unit areaTFeasible solution CT-FS, formula is as follows
If the error of feasible solution and optimal solution is very small, so that it may approximately replace optimal solution with feasible solution.This model
Optimal solution is difficult to obtain, therefore can not directly determine error, but present invention discover that the lower bound of optimal solution is easier to obtain, because
This may infer that the error of feasible solution and optimal solution is also very small if the error of the lower bound of feasible solution and optimal solution is very small, this
When feasible solution can be with approximate substitution optimal solution.
2. optimal solution lower bound
To sum up, the lower bound of optimal solution will be found in this part.It is known that unit area freight includes backhaul expense and backhaul
Expense solves optimal solution lower bound to this two parts respectively, then aggregation summation, that is, obtains the lower bound of unit area freight.
From fig. 5, it can be seen that the optimal solution of backhaul expense meets following two condition.
jr-1(x) and jr(x) adjacent, for
The average distance for meeting first condition is β0A1/2；Meet second condition, from facility jr-1(x) facility j is arrivedr(x)
Average distance be
Accordingly, backhaul expense is obtainedOptimal solutionFormula
When due to backhaul, client directly returns to the departure place of client from facility.Therefore, when client accesses r grade facility
When being the close facility of its r, the optimal solution of backhaul expense can be obtained.γrA1/2Indicate that client returns to from the facility of r grade
The average distance of its departure place, γrFor fromrThe facility of grade returns to the distance proportion coefficient of its departure place, γRFor from R grade
Facility return to the distance proportion coefficient of its departure place, γrNumerical value can be to βrIt is ranked up and obtains.
Accordingly, the backhaul expense of unit area is obtainedOptimal solutionExpression formula,
According to formula (20) and (21), the lower bound C of the freight of unit area can be obtainedT-LB, formula is
3. feasible solution and optimal solution lower bound error
Next, the feasible solution C of the freight of unit of account areaT-FSWith lower bound CT-LBError.Formula is as follows
According to formula (19) and (22), can be obtained
It according to formula (24), is computed, obtains error (gap) with the variation of R and q, Fig. 6 is respectively shown in q=0.1
When the trend that changes with error when the R trend changed and R=5 with q of error.
In Fig. 6 when q=0.1, when loss probability is constant, although error is increased with the increase of R, maximum value
It is less than 0.003%；In Fig. 6 when R=5, with the increase of loss probability q, then error, which first increases, to be reduced, but maximum value is not yet
More than 1.2%.
In Practical Project, facility probability of damage does not all exceed 0.5 generally, because excessively high damage probability means that its is non-
Normal unreliable unsuitable construction, herein under the premise of, it can be seen that the error amount of this model is less than 1%.To sum up, the present invention can
Approximately to replace optimal solution come the freight of unit of account area with feasible solution.
4. model solution
According to formula (10), the total cost of the unit area of homogeneity continuous level problem, formula is can be obtained in (12) and (19)
It is as follows
It can be seen that homogeneity continuous level problem only one decision variable A from the formula and the function be unimodal letter
Number.Therefore, it can be enabled by carrying out derivation to formula (25)Obtain optimal A*, and then obtain optimal unit
Total cost C (the A of area*)。
4.2, heterogeneous continuous level location problem
Assuming that parameters f (x), λ (x) in the heterogeneous continuous level S of two dimension,Variation with q (x) is relatively slow
Slowly, the area continuous function of initial service area of the Facilities Construction at the x of positionIt indicates, is replaced with A (x)
In generation, finds the position collection x of discrete facility, constructs total cost function.
Assuming that facility is dense distribution, then the area of two-dimentional heterogeneous continuous level S is greater than the first of any facility from far away
The area of beginning coverage, i.e.,Therefore, boundary can be ignored under the premise of not influencing total cost
It is heterogeneous；When model parameter is approximate constant in certain region, the area A (x) in initial effects region is in the area
It should be approximate constant.Based on the above analysis, any position x ∈ S and its neighborhood are calculated using formula (25), and (approximation is regarded as
Homogeneity continuous level) at unit area total cost.
In view of the parameter setting of heterogeneous continuous level, indicate that the total of unit area near x is taken using C (x, A (x))
With formula is as follows
Formula (26) can be solved, obtain service area A optimal at x*(x) function.And then using integral
Method, approximately obtains the optimal total cost of limited heterogeneous continuous level S, and formula is as follows
C*=∫x∈SC(x,A*(x))dx (27)
Due to optimal service area A*(x) inverse reflects the facility density near x, therefore using the method for integral
Optimal Facilities Construction quantity can be obtained, formula is as follows
N*≈∫x∈S[A*(x)]-1dx (28)
The optimal service area A acquired at this time*(x) and optimal Facilities Construction number N*Reality can not be directly applied to, because
Optimum results to need in reality are the installation location of node and the access order of client, it is still necessary to by model result carry out from
Dispersion can just obtain optimum results.
4.3, model structure discretization
The specific solution throughway of the discretization algorithm is as follows, and process is as shown in Figure 9:
Planning region PS, PS ∈ S are placed in the first quartile of coordinate system by the first step, then the coordinate of every bit is available
(x, y) is indicated.In general, the boundary of planning region is curve, the size and Orientation of boundary repulsive force for ease of calculation,
The present invention approximatively uses one group of straight line to be expressed as chimb circle, as shown in Figure 7.Its group of functions uses y=-ax+c, wherein a table
Show that coefficient vector, arbitrary element are indicated with a, c indicates constant vector, and arbitrary element is indicated with c.
Second step will be that cellular, size are w × w one by one by the surrounded region division of chimb circle, and center is sat
Mark uses (xi,yi) indicate, wherein i=1,2,3 ..., I indicate the number of cellular.For any cellular i, its customer service is calculated
Radius g (xi,yi), formula is as follows
Third step randomly selects NC cellular in planning region PS, its center is enabled to indicate the position of node addressing,
The number of node is indicated using nc=1,2,3 ..., NC, nj=1,2,3 ..., NC, i.e. cellular nc, nj indicates the member comprising node
Born of the same parents.Enable the number of iterations m=1.
4th step calculates between each cellular and between cellular and boundary for each cellular for being chosen for node
Repulsive force.
Similar with the Social Dynamics thought in pedestrian simulation research, there is also classes for each node in node siting analysis
As repulsive force effect.This is because if being overlapped when the coverage of any two node has coincidence phenomenon
The client in region just can only select one of node to receive service, to cause the missing in two node serve regions.In order to
Will be far to avoid there is the phenomenon that coverage coincidence in the phenomenon that avoiding client from lacking, the positions of two nodes, this is also
Illustrate two similar nodes there is repulsive force, size can be reduced with the increase of two nodal distances until disappearing.
Therefore the effect that repulsive force is introduced between adjacent cellular, for determining the movement of node location.
Therefore, any cellular nc is by any other repulsive force h comprising node cellular njRFThe setting of (nc, nj) size are as follows:
Wherein h indicates maximum repulsive force, lnc-njIndicate cellular nc and comprising node cellular nj the distance between, formula
For
It is possible thereby to respectively obtain the repulsive force in the component of x-axis and y-axis direction
In order to guarantee that cellular nc is not appeared in except chimb circle, Arbitrary Boundaries cellular equally exists repulsion to cellular nc
Power, size use following formula
hB=(NC+1) h (34)
It is possible thereby to respectively obtain Arbitrary Boundaries repulsive force to cellular nc x-axis and y-axis direction component
By summing to cellular nc in all component of x-axis and y-axis direction, it is available its in x-axis and y-axis direction institute
The resultant force size received, uses H respectivelyx(nc) and Hy(nc) it indicates.The stress size of all nodes is calculated using aforesaid way.If
The stress of all nodes is 0, goes to the 7th step；Otherwise the 5th step is carried out.
5th step, according to the direction of resultant force suffered by cellular nc resultant force size calculate node suffered by x-axis and y-axis direction,
Judge direction closest in 8 directions around the direction and cellular nc, chooses cellular adjacent in this direction as section
The mobile target cellular of point, the rule are as shown in Figure 8.In fig. 8, black cellular indicates the cellular where node, and grey indicates
8 neighbours' cellulars of black cellular, dotted line indicate 8 directions of black cellular and neighbours' cellular, and solid line indicates all rows of node
The resultant direction of repulsion, by evolution rule it is found that the cellular in the black cellular upper right corner is the node target cellular to be moved.
6th step, the node motion that all stress are not zero is into new target cellular, m=m+1.If m < mmax, weight
It is new to carry out the 4th step；Otherwise, third step is re-started.
7th step using cellular center as node addressing point, and uses Thiessen polygon according to obtained cellular position
Method obtains the service area boundary of each node.
By taking Beijing Tianjin and Hebei Region transportation network as an example, the beneficial effect of this method is inquired into.
1, data source
The CRLP-IITT model that the present invention constructs is continuous model, and used parametric form needs to be successive value, this
Engineering reality is put and is not met, therefore the present invention first obtains corresponding original discrete data, is just applied to after processing stage
This model.
County-level city of the original discrete data in Beijing Tianjin and Hebei Region, with the administrative center position of each county-level city
As the alternative point set of node, be worth based on the size of population of various regions, obtained after processing the location point customer service requirement amount,
The probability that the node construction cost and node failure of the location point occur.
Specifically, the present embodiment data are made of Beijing, Tianjin, the regional counties and cities' data in Hebei three, with Beijing Tianjin and Hebei Region
Interior county-level city is as discrete standby addressing point, using the ratio value of the size of population as the demand of client, and it is basic herein
The upper construction cost and loss probability for obtaining facility.
It altogether include 204 standby addressing points in embodiment, comprising: the coordinate points in 16 areas, Beijing are (where administrative center
Ground), population (ten thousand people) and area (km2)；Coordinate points (administrative center location), population and the face in 16 districts in Tianjin
Product；Coordinate points (administrative center location), population and the area of 172 county-level cities, Hebei province.Above data is respectively derived from
Beijing, census datas in 2010 of Tianjin and administrative area division and Hebei province's demographic datas in 2009 and administrative area
It divides.Figure 10 is distribution situation figure of 204 standby addressing points in Beijing Tianjin and Hebei Region.
It should be noted that each point includes several attributes in the data: geographical coordinate point is both the alternative position of node
It sets a little, is also the homeposition point of client；Each location point has respective construction cost, probability of damage and the client's
Demand for services amount.
These data are the original discrete datas of the present embodiment, it is still necessary to handle, are obtained continuously by certain smoothing method
The parametric function that model needs.
2, data processing
Data are set first, then data are smoothed, obtain the continuous parameter of continuous model needs
Function.Specific roadmap is as follows:
(1) initial data is set
Assuming that customer demand and the local size of population are positively correlated, i.e., every hundred people generates a demand, corresponding four groups of numbers
The size of population (ten thousand people) in is multiplied by 102Obtain the customer demand at each demand pointAccording to client's need of each counties and districts
The amount of asking determines the fixed cost of current counties and districts construction node, i.e.,The damage of each alternate node is general
Rate assumes there is negatively correlated relationship with the construction cost of node, enables its expression formula be in the present inventionIts
Middle ρ is probability of damage coefficient, which mainly considers that its corresponding node of the node big for construction cost is reliably just high,
It is more difficult by damage；The transportation cost of unit demandWith client LxTo position ZxDistance it is related, phase relation
Number, that is, freight rates are set as ω, therefore unit transportation cost can be simply with freight rates ω multiplied by transportation rangeIt obtains
?.
Transportation range is the transportation route length between each position, since the obtained data of the present invention are each alternatively to put
Latitude coordinates, so will be calculated using great-circle distance formula.If A point longitude and latitude is (LonA, LatA), B point longitude and latitude is
(LonB, LatB), then distance between two points formula are as follows:
C=sin (LatA) * sin (LatB) * cos (LonA-LonB)+cos (MLatA) * cos (LatB)
Distance=R*Arccos (C) * Pi/180
Or the longitude and latitude of two o'clock directly can be inputted into certain calculating network address, great-circle distance value can be directly obtained.
Simultaneously according to the research of Qureshi et al., point-to-point transmission has the scene of transit route smaller in reality, through studying
After verifying, it is believed that increasing distance coefficient 1.2 is that comparison is reasonable, therefore the transportation range of point-to-point transmission is 1.2 times in the present invention
Great-circle distance between two places.
(2) discrete data serialization
Data sequentialisation refers to, on the basis of the original discrete data of acquisition, is converted to and meets continuous model requirement
Functional form.
The spherical coordinate data of discrete data are changed into the plane coordinates data in plane coordinate system first, then basis
Discrete plane coordinates data pass through the planning region of matlab software building continuous level, discrete in last Continuum Model
Data, the probability of damage including customer demand, the construction cost of node and node.
Broken line in Figure 11 illustrates the approximate planning region boundary of Beijing Tianjin and Hebei Region.
It is continuous since the demand of client is to carry out summation by the demand in region to be abstracted into some acquisition
The demand metric density at arbitrary point should be obtained when change, and node construction cost and node probability of damage are then due to fixed point
Position determine, therefore the numerical value at arbitrary point should be obtained when serialization.
From there through matlab software to discrete customer demand metric density, node construction cost and node probability of damage
Serialization is carried out using the method for interpolation, and the data of serialization are smoothed and obtain final continuous data, respectively
For λ (x), f (x) and q (x).The punishment cost for not obtaining service for client sets it in planning region at arbitrary point herein
All be it is fixed and identical, i.e.,
(3) parameter setting
Parameter default setting in CRLP-IITT model is as follows: α=1, α indicate facility cost coefficient, ρ=0.1, ω
=1,And R=5.
The present invention is shown the result of data processing under conditions of default parameter value.Figure 12 is customer demand
Density profile, it can be seen that customer demand is concentrated mainly on the cities such as Beijing, Tianjin, Qinhuangdao, Baoding, Shijiazhuang, Handan,
This is consistent with discrete data, also illustrates that the serialization of discrete data is feasible.
Figure 13 and Figure 14 then respectively shows the distribution of the construction cost and probability of damage of node in continuous planning region.
3, model is verified
There are two unique hypothesis for CRLP-IITT model: information failure increases return expense, and the present invention will lead to first
The comparative analysis for crossing optimum results, illustrate the two hypothesis importance and necessities, and verify this model necessity and
Feasibility.
(1) information is effectively compared with optimum results under information failure scenario
In existing research, the objective function of the fully effective scene lower node location optimization model of information is
It is input with Beijing Tianjin and Hebei Region transportation network data source, while to the fully effective situation of information and CRLP-IITT mould
The objective function of type can embody information and effectively optimize the difference of layout under two different scenes of failure.Table 1 is to join substantially
Under the same conditions, with the variation of loss probability coefficient ρ, the optimum results that two group models obtain are compared for number setting.
Effectively and under information failure scenario two group model optimum results compare 1 information of table
The 1 different meaning of parameters of middle part dtex of table is as follows:
Under information failure scenario, C is used*Indicate optimal system total cost, optimal number of nodes is N*, optimal node serve
Area is A*(x)；Under information is effective, optimal system total cost is usedIt indicates, optimal number of nodes isOptimal node
Service areaAt this point, using CPI-IIIndicate use information effectively under optimal solution when, i.e., withFor node serve
When area, if client accesses according to the access strategy under information failure scenario, and system total cost when service is finally obtained.Indicate the difference degree of information failure and the system total cost in the effective situation of information,Table
Show under information failure conditions, optimal case A*(x) with non-optimal schemeSystem total cost difference degree.
Compare the total cost C in different situation drags*With number of nodes N*Difference, it can be found that information failure situation under
Optimal total cost C*With number of nodes N*The optimal total cost being all larger than in the effective situation of informationAnd number of nodesCorrespondingly,
Difference degree εPIAlso constantly expand in the increase with probability of damage coefficient, minimum value is not less than 10%.
This illustrate the degree of mobility of information for node Quantity and system synthesis this influence still obviously,
And its influence degree is also increasingly severe with the increase of probability of damage.This also illustrates only have when client can obtain information
When effect, and minimum node Quantity and total cost could be obtained based on the progress node addressing of this condition.Therefore, information skill
One important goal of art development is ensuring that client can obtain various reliable information in real time and can make accordingly most
Suitable decision.But in the case of this information is effective in even following one section of long time for current by
In the external factor influence in various inherences so that can not fully achieve, it is also formed the status of information failure.
If effectively assuming to carry out node addressing based on information, the status of information failure can make the total expense actually occurred
With the total cost predicted when being greater than and planning, the C being embodied in table 1PI-IIIt is greater thanIn addition, also observing total cost CPI-II
Total cost C also greater than under information failure conditions*, difference degree εPI-IIAlso with probability of damage increase and increase.Although
Difference degree is simultaneously little, but still illustrates that unreasonable addressing scheme will cause the increase of system total cost.
(2) consider and do not consider that optimum results compare when return expense
In order to embody the importance of return expense, the present invention adapts the program of establishment, has obtained one and has not examined
Consider program when return expense, it is to consider when inputting and do not consider return expense that table 2, which is with Beijing Tianjin and Hebei Region transportation network data source,
The optimum results comparison that two programs of used time obtain.
The costimating of continuous model under 2 return expense presence or absence of table
The 2 different meaning of parameters of middle part dtex of table is as follows:
When not considering return expense, optimal system total cost is usedIt indicates, optimal number of nodes isIt is optimal
Node serve area Indicate that system is total in the case of considering return expense and not considering return expense
The difference degree of expense,Indicate the optimal case A when considering return expense*(x) with it is non-optimal
SchemeSystem total cost difference degree.
Table 2 does not consider the system total cost of return expense from a visual impression is numerically demonstratedIt is obvious small
In the system total cost C for considering return expense*, and difference degree εNIBIt is gradually reduced with the increase of node damage probability, but
Its difference degree is still below 50%.
Do not consider return expense when carrying out node addressing, then actually occur there are the total cost C of return expenseNIB-IB
Not only it is higher than the optimal total cost estimatedAnd it is higher than optimal node addressing total cost C when considering return expense*,
Difference degree εNIB-IBLikewise as node probability of damage increase and reduce.This illustrate the reliability effect of node return expense
The occupied ratio in total cost.More reliable node, in the effective situation of information, client can more access least node
To service, in turn result in return expense occupy in total cost shared by ratio it is higher.Conversely, the ratio shared by it is got over
It is low.Unreasonable node addressing does not consider the node addressing of return expense, will cause that the expense of estimating is too low and and actual cost
Have big difference.Thus the consequence caused by, gently then system benefit reduces.It is serious or even will cause if infusion of financial resources is insufficient
The paralysis of whole system.
Conclusions, which demonstrate, considers that limited information scene, increase return cost are all very important, and illustrates the two
The importance and necessity of hypothesis has apparent adjustment and optimization to the optimum results of addressing, to this progress of system synthesis
Optimization, demonstrates the Necessity and feasibility of this model.
Key problem in technology point of the invention:
(1) access strategy description of the client under information failure scenario in transportation network
It is failed based on information it is assumed that the characteristics of being usually present return cost according to movement operation, the invention proposes one
The completely new client's access strategy of kind.
No matter in any case, whether transit node fails, client can not learn node realtime running state, he can only
One group of transit node for accessing designated order one by one, is serviced and is returned Original Departure Point, or seek all over after searching out enabled node
It returns to one's starting point after having no result and receives rejection penalty.Transit node position and access order are all according to expection failure probability and are
System total cost determines.
In this similar to trial and error process access strategy, under any situation, each client only can be according to true in advance
Fixed access order successively accesses specified transit node, and can rest on the node of first available transportation service, or
Person visits all over have no result after, abandon transportation service and receive rejection penalty.
(2) building of the region transportation network reliability addressing continuous model under information failure scenario
It is described based on the access strategy for seeking service to client in the region transportation network under information failure scenario, the present invention
Construct the reliability Continuous Location of transit node, it is therefore intended that: any position is obtained in extensive two-dimentional addressing region
The optimal facilities services region area at the place of setting obtains optimal Facilities Construction quantity in turn, thus in view of being likely to occur
In the case where, it is intended that the total expected cost of the system minimized.The decision variable of this model not just determines construction transport
The access order of each client is also predefined in the position of node and quantity, to obtain optimal target value.
In model construction process, find original continuous model can not direct solution, therefore done with continuous approximation
Method is converted: being considered the continuous level problem of homogeneity first, is extended to heterogeneous continuous level after discovery rule, finally
It has obtained to solve, has also complied with actual CRLP-IITT model.
(3) solution of the region transportation network reliability addressing continuous model under information failure scenario
The optimum results of CRLP-IITT model are optimal service region and the construction number of transit node in the transportation network of region
Amount, is not directly applicable engineering practice, needs to carry out sliding-model control to result, thus obtain include node installation location with
The prioritization scheme of client's access order.
(4) present invention utilizes the thought based on cellular automata, the characteristics of for CRLP-IITT model, has customized and has been applicable in
In discretization algorithm of the invention, and utilize Matlab software programming respective code.
Abbreviation, English and Key Term definition:
1, CRLP-IITT model: English is Continuous reliable location problem with
Imperfect information and two-way trips, that is, consider under the limited information scene that the present invention constructs two-way
The reliability Continuous Location of stroke.
The content being not described in detail in this specification belongs to the prior art well known to professional and technical personnel in the field.
Claims (7)
1. considering the continuous site selecting method of reliability of return under a kind of limited information scene, which is characterized in that including following step
It is rapid:
Step 1, site selection model are assumed
It (1) is not that permanently can be used, there are certain probability of damage after transit node construction is come into operation；
(2) probability of damage is independent from each other between each transit node；
(3) client only knows the initial information of transit node, in this case it is not apparent that the real time information of transit node；
(4) client will successively access prevailing traffic node and spare transit node according to assignment order, and can only access distribution
To the transit node of client oneself, regardless of whether being serviced, other transit nodes will not be all visited again；
(5) if client finally there is no service, then client can only abandon and receive certain rejection penalty；
(6) no matter whether client is serviced, and will finally return to Original Departure Point, that is, there are return costs；
Step 2, defined variable
In transportation service region, the heterogeneous continuous level of two dimension is set as S, and arbitrary point position is set as x, client's fortune at the x of position
Defeated demand for services λ (x),It indicates；Transit node is built at the arbitrary point position x in the transportation service region,
Fixed construction cost is f (x), and the probability of damage for the transit node built at the x of position is q (x), and client is put at the x of position
Abandoning the punishment cost that transportation service obtains isThe grade that client accesses certain transit node is r, and each client accesses transport section
The maximum access number of point is defined as R, with client LxIndicate the client on the x of position, transit node ZxIt indicates on the x of position
Node；
Step 3, site selection model building
3.1, the site selection model expression formula under unit area is solved in homogeneity continuous level；
Homogeneity continuous level is a unlimited two-dimensional space, the related parameter values in homogeneity continuous level, on all positions
It is identical and fixes, then f (x)=f, λ (x)=λ,Wherein f indicates transit node
The constant of construction cost, λ indicate the constant of client's transportation service demand,Indicate that the constant of punishment cost, q indicate transit node
The constant of probability of damage；
(1) analyzed area is determined
Assuming that the initial coverage of each transit node is the mesh space being made of regular hexagon, and area A, and it is every
One facility is all located at the center of regular hexagon, and regular hexagon is divided into 12 equal small triangles, it is small to choose one of them
Triangle is as analyzed areaAnalyzed areaArea be
(2) the site selection model expression formula under unit area is determined
1. solving the Facilities Construction cost C under unit areaF；
2. solving the punishment cost C under unit areaP；
3. solving the transportation cost C under unit areaT；
To the Facilities Construction cost C under unit areaF, punishment cost C under unit areaPWith the transportation cost under unit area
CTSummation, obtains the site selection model expression formula under unit area；
3.2, the site selection model expression formula in the heterogeneous continuous level S of two dimension is solved according to 3.1 result, finally obtained in two dimension
The optimal total cost of site selection model and optimal Facilities Construction quantity in heterogeneous continuous level；
3.3, discretization is carried out to the optimal total cost of site selection model and optimal Facilities Construction quantity, obtains the optimization of site selection model
As a result.
2. considering the continuous site selecting method of reliability of return under limited information scene as described in claim 1, feature exists
In Facilities Construction cost C under unit areaFCalculation formula it is as follows:
CF=f/A (10).
3. considering the continuous site selecting method of reliability of return under limited information scene as claimed in claim 2, feature exists
In punishment cost C under unit areaPCalculation formula it is as follows:
In formula, qR+1Represent the probability that client in unit area does not obtain service.
4. considering the continuous site selecting method of reliability of return under limited information scene as claimed in claim 3, feature exists
In transportation cost C under unit areaTWith feasible solution CT-FSIt indicates, calculation formula is as follows:
In formula, ω is freight rates, wherein βrFor from the facility of r grade to analyzed areaDistance proportion coefficient, β0For from
The facility of 0 grade is to analyzed areaDistance proportion coefficient, βRFor from the facility of R grade to analyzed areaDistance proportion
Coefficient.
5. considering the continuous site selecting method of reliability of return under limited information scene as claimed in claim 4, feature exists
In the site selection model expression formula under unit area is as follows:
In formula, C (A) represents the total cost of unit area in homogeneity continuous level.
6. considering the continuous site selecting method of reliability of return under limited information scene as claimed in claim 5, feature exists
In, parameters f (x), the λ (x) in the heterogeneous continuous level S of two dimension are assumed in step 3.2,Variation with q (x) compares
Relatively slowly, the area continuous function of initial service area of the Facilities Construction at the x of positionX ∈ S indicates, then two
The expression formula for tieing up site selection model in heterogeneous continuous level is as follows:
In formula, C (x, A (x)) represents the total cost of unit area in the heterogeneous continuous level of two dimension；
To formula (26) derivation, service area function A optimal at x is obtained*(x), two dimension is then obtained using the method for integral
The optimal total cost C of heterogeneous continuous level S*With optimal Facilities Construction quantity N*, calculation formula is as follows:
C*=∫x∈SC(x,A*(x))dx (27)
N*≈∫x∈S[A*(x)]-1dx (28)。
7. considering the continuous site selecting method of reliability of return under limited information scene as claimed in claim 6, feature exists
In the process of discretization algorithm is as follows:
3.3.1, planning region PS, PS ∈ S are initialized as chimb circle first；
3.3.2, the region division for surrounding chimb circle is multiple cellulars, and the size of cellular is w × w, and the centre coordinate of cellular is used
(xi,yi) indicate, wherein i=1,2,3 ..., I indicate that the number of cellular calculates the service radius g of client for any cellular i
(xi,yi), formula is as follows:
A in formula*(xi,yi) indicate in (xi,yi) at optimal service area function；
3.3.3, NC cellular is randomly selected, the center of cellular is enabled to indicate the position of node addressing, using nc=1,2,
3 ..., NC, nj=1,2,3 ..., NC indicate the number of node, i.e. cellular nc, nj indicates the cellular comprising node, enable the number of iterations m
=1；
3.3.4, the repulsive force between each cellular comprising node and the cellular comprising node and the row between boundary are calculated
Repulsion；
Cellular nc comprising node is h by the repulsive force of the cellular nj comprising nodeRF(nc, nj), calculation formula is as follows:
Wherein h indicates maximum repulsive force, lnc-njIndicate the cellular nc comprising node and between cellular nj comprising node away from
From formula are as follows:
hRFThe component of (nc, nj) in x-axis and y-axis direction is respectively as follows:
Cellular nc comprising node and the repulsive force between boundary are hB, calculation formula is as follows:
hB=(NC+1) h (34)
hBComponent in x-axis and y-axis direction is respectively as follows:
All component to the cellular comprising node in x-axis and y-axis direction are summed, and are obtained suffered by the x-axis and y-axis direction
Resultant force size, uses H respectivelyx(nc) and Hy(nc) it indicates, if all cellular stress comprising node are 0, goes to step
3.3.7；Otherwise step 3.3.5 is carried out；
3.3.5, according to the resultant force size suffered by x-axis and y-axis direction of the cellular comprising node, the cellular institute comprising node is calculated
It is found out and suffered resultant force by the direction of resultant force according to 8 directions in the direction of suffered resultant force and the cellular surrounding comprising node
Target element of the cellular as the cellular movement comprising node on closest direction is chosen in the closest direction in direction
Born of the same parents；
3.3.6, the cellular comprising node that all stress are not zero is moved in corresponding target cellular, m=m+1；
If m < mmax, re-start step 3.3.4；Otherwise, step 3.3.3 is re-started；
3.3.7, according to the obtained cellular position comprising node, the service of each node is obtained using Thiessen polygon method
Zone boundary.
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