CN108376105B - Radar network task allocation method based on time-effect constraint combination bilateral auction - Google Patents

Radar network task allocation method based on time-effect constraint combination bilateral auction Download PDF

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CN108376105B
CN108376105B CN201810249975.6A CN201810249975A CN108376105B CN 108376105 B CN108376105 B CN 108376105B CN 201810249975 A CN201810249975 A CN 201810249975A CN 108376105 B CN108376105 B CN 108376105B
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张天贤
田团伟
徐龙潇
李固冲
孔令讲
崔国龙
易伟
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University of Electronic Science and Technology of China
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Abstract

The invention discloses a radar network task allocation method based on an aging constraint combination bilateral auction, belongs to the technical field of radars, and relates to an aging constraint combination bilateral auction technology for multifunctional radar network task allocation. The invention overcomes the problems that the task value changes along with the time, and the radar resource can not be effectively distributed when the radar can not process the task in real time. The method is characterized in that demand information of task demanders and supply information of a radar are uploaded to an auction center, then the auction center determines prices of various resources according to supply-demand relations of the radar resources, finally determines distribution sequences of the tasks according to task arrival time of the tasks, and distributes various resources of a plurality of tasks by taking total profit maximization as a distribution criterion.

Description

Radar network task allocation method based on time-effect constraint combination bilateral auction
Technical Field
The invention belongs to the technical field of radars, and relates to the technical research of timeliness constraint combination bilateral auction of task allocation of a multifunctional radar network.
Background
The multifunctional radar network task allocation means that different types of resources such as limited time, antennas, signal processing units and the like are reasonably allocated to searching, tracking, identifying and other types of tasks according to the real-time situation of a battlefield. The resource requirements of the tasks to be executed can be dynamically influenced by the high-speed and high-maneuverability characteristics of the target, enemy interference and the like, and the resource allocation amount of each task to be executed needs to be changed in real time in different time periods. Reasonable resource allocation can improve the overall operational efficiency of the multifunctional radar network, and is one of the research hotspots of experts in the field of radars at home and abroad at present.
For the multifunctional radar network system, the distribution of limited radar resources to different tasks is an NP-hard problem, and the optimal distribution is difficult to solve. The combined bilateral auction method combines the advantages of the combined auction method and the bilateral auction method, and can effectively solve the NP-hard problem of multifunctional radar network resource allocation. In the documents "phase array radio resource management using connected double auctions, IEEE Transactions on aeronautics and Electronic Systems, vol.51, No.3, pp.2212-2224,2015", the multi-function radar tasks are distributed by using the combined double auction method, but the time-varying property of the task value and the response time of the radar are not considered. From the published literature at present, the distribution of multifunctional radar network tasks with time-varying task values using a combined bilateral auction method has not been studied.
Disclosure of Invention
The invention aims to research and design a multifunctional radar network task allocation method based on time-effect constraint combination bilateral auction aiming at the problems in the background art, and solves the problems that the task value changes along with the time lapse, and the task cannot be effectively allocated when the radar cannot process the task in real time.
The method comprises the steps of uploading demand information of task demanders and supply information of a radar to an auction center, determining prices of various resources by the auction center according to supply-demand relations of the radar resources, determining distribution sequences of the tasks according to task arrival time of the tasks, and distributing a plurality of tasks and various resources by taking total profit maximization as a distribution criterion.
For the convenience of describing the contents of the present invention, the following terms are first explained:
the term 1: auction center
The auction center is responsible for collecting the demand information of all tasks and the supply information of all radars and distributing the radar resources according to the price mechanism
The term 2: task deadline
Task deadline TdI.e. the deadline by which the task is executed, is 0 after this time, and needs to be discarded
The term 3: response time
Response time TrI.e. the time when the radar starts to perform a task
The term 4: price of resources
The resource price is the price of a unit resource, which is determined by the resource demand and supply, and the resource prices of different tasks are different.
The invention provides a radar network task allocation method based on time-effect constraint combination bilateral auction, which comprises the following steps:
step 1: collecting demand and supply information;
the auction center collects M task requesters D ═ D1,d2,…,dM]Requirement information of { quantity X of required resources, resource type L, task deadline TdR and N radars R ═ R1,r2,…,rN]Supply information { amount of supply resources Y, resource type L, response time Tr}；
Step 2: determining the price of the resource;
for each task, the auction center determines the price of the resource according to the supply-demand relationship of the radar resource, namely the price of the l-th resource of the ith task is determined by the difference value of the required quantity of the l-th resource of the ith task and the total supply quantity of all radars to the l-th resource;
and step 3: solving the system income;
the difference value between the total income of all tasks and the cost consumed by the radar for providing resources is finished to obtain the total income of the system, the maximization of the system income is taken as an allocation criterion, and when the system income reaches the maximum value, the task is indicated to be optimally allocated; for diI-1, 2, …, M, the task needs to be completed before its task arrival time, only the response time TrN less than its cut-off timecThe i part of radars can execute the tasks; as the task value is reduced along with the time, the response time of different radars is different, so the gains obtained by the execution of different radars are different; in order to maximize the system benefit when the ith task is completed, the selected radars are sequenced according to the sequence of the resource consumption cost from low to high, namely, the radars with low resource consumption are preferentially used for distributing the tasks, and the sequenced radars from high to low are collected into
Figure GDA0003263385200000021
And 4, step 4: distributing tasks;
step 4.1: determining a task allocation sequence;
sequencing all task demanders according to the sequence of the task cut-off time from low to high, namely preferentially executing the task with the task cut-off time being earlier;
step 4.2: judging whether the task can be successfully executed;
determining task demander diNumber of demands on class I resource xilAnd the sum of the number of resources provided by the selected radar
Figure GDA0003263385200000031
Wherein L is 1,2, …, L; if it is
Figure GDA0003263385200000032
Namely, the demand is larger than the supply quantity, the task failure is judged, and the number M of the task failure is calculatedfail Adding 1, turning to step 4.4, otherwise, turning to step 4.5;
step 4.3: the task completion rate is constant;
number of failures in task MfailIf the response time T is more than 0, the response time T of the radar is adjustedr- δ, where δ denotes the adjustment step size determined according to the actual situation, increasing the chance of radar selection, go to step 4.2, reselect the selection response time TrRadar less than its arrival time;
step 4.4: a task allocation process;
determining task demander diNumber of demands on class I resource xilWhether or not greater than radar r1Number y that can be provided1lIf x isil≤y1lThen, indicate diTask of (1) is represented by1Finish, update radar r1Resource information of y1l-xilGo to the next class of resource, otherwise indicate r1Cannot satisfy d aloneiD is updated if the demand for the l-th resource is satisfiediIs xil-y1lAnd steering radar r2；
Determining task demander diThe remaining demand resource of (2) and radar r2The size of the number of resources that can be provided, if (x)il-y1l)≤y2lThen, indicate diThe demand allocation of the l-type resource is completed, and the radar r is updated1And r2The resource information of (d) is allocated to the next kind of resource, otherwise, the resource information of (d) is updatediIs xil-y1l-y2lIs turned to r3(ii) a And so on until d is finishediThe distribution of the l type resource is turned to the distribution of the next type resource; repeat step 4.5 until d is completeiSuccessful allocation of all types of resources indicates diThe task is successfully distributed, and the next task is turned to; and repeating the steps 4.2-4.5 until the distribution of all tasks is completed.
Further, in the step 2, the price p of the l-th type resource of the i-th task is determined by adopting the following formulail；
Figure GDA0003263385200000033
Wherein, lambda is an adjusting factor influenced by the supply and demand difference quantity, and is determined according to the actual situation, and xilRepresenting the required quantity of the ith task to the l-th type resource, yjlIndicating the supply quantity of the jth radar to the l-th type resource.
Further, the specific method of the step 3 is;
step 3.1: determining a task value;
for the ith task, i ═ 1,2, …, M, the response time T is selectedrLess than its cut-off time
Figure GDA0003263385200000041
The radar performs its task; average value of its L resources
Figure GDA0003263385200000042
Comprises the following steps:
Figure GDA0003263385200000043
wherein: t isi dIndicates the time-to-arrival of the ith task,
Figure GDA0003263385200000044
denotes the response time of the j-th radar, pilThe price of the l type resource of the ith task is represented;
step 3.2: determining a resource consumption cost;
for the j-th radar rjWhere j is 1,2, …, N, the average cost per unit resource amount of L resources consumed in executing a task, i.e., the cost, is determined based on the task start time and the supply price
Figure GDA0003263385200000045
Step 3.3: the total profit of the system is maximized;
the gain in completing the ith task is the product of its demand for all resources and the average value of the L resources, i.e.
Figure GDA0003263385200000046
Radar rjThe cost consumed is the product of the supply amount of the supply resource and the average cost per unit resource amount of the L kinds of resources, i.e. the cost
Figure GDA0003263385200000047
The system gain obtained by completing the ith task is
Figure GDA0003263385200000048
To maximize the system gain J in completing the ith taskiWill be selected
Figure GDA0003263385200000049
The radars are sorted according to the sequence of low resource consumption cost to high resource consumption cost, namely, the radars with low resource consumption are preferentially used for distributing tasks, and the sorted radar set is
Figure GDA00032633852000000410
The invention has the beneficial effects that: the method comprises the steps of uploading demand information of task demanders and supply information of a radar to an auction center, determining prices of various resources by the auction center according to supply-demand relations of the radar resources, determining distribution sequences of the tasks according to task arrival time of the tasks, and distributing various resources of a plurality of tasks by taking total profit maximization as a distribution criterion. The method has the advantages that the method is suitable for the actual situation that the task value changes along with time and the radar cannot process the task in real time, and simultaneously performs optimal distribution on multi-task and multi-resource, thereby improving the combat efficiency of the multifunctional radar network. The invention can be applied to the fields of civil military and the like.
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FIG. 1 is a system block diagram of a method provided by the present invention;
FIG. 2 is a block flow diagram of a method provided by the present invention;
FIG. 3 is a schematic view of resource prices determined for various tasks in accordance with an embodiment of the present invention;
FIG. 4 is a diagram illustrating task completion rates in accordance with an embodiment of the present invention;
FIG. 5 is a schematic diagram of the total revenue of the system in accordance with an embodiment of the present invention.
Detailed Description
The invention mainly adopts a simulation experiment method for verification, and all the steps and conclusions are verified to be correct on Matlab 2015. The present invention will be described in further detail with reference to specific embodiments.
Step 1: collecting demand and supply information
The auction center collects M task requesters D ═ D1,d2,…,dM]Requirement information of { quantity X of required resources, resource type L, task deadline TdR and N radars R ═ R1,r2,…,rN]Supply information { amount of supply resources Y, resource type L, response time Tr}。
The following matrix representation is used:
Figure GDA0003263385200000051
xilrepresenting the required quantity of the ith task to the l-th type resource;
wherein i is 1,2, …, M, L is 1,2, …, L;
Figure GDA0003263385200000052
Ti drepresents the intercept time of the ith task, where i＝1,2,…,M；
Figure GDA0003263385200000053
yjlRepresenting the supply quantity of the jth radar to the l-type resource;
wherein j is 1,2, …, N, L is 1,2, …, L;
Figure GDA0003263385200000055
Figure GDA0003263385200000054
represents the response time of the j-th radar, wherein j is 1,2, …, N;
step 2: determining resource prices
Aiming at each task, the auction center determines the price of the l-th resource of the ith task according to the supply and demand relationship of radar resources, namely the difference value of the required quantity of the ith task to the l-th resource and the total supply quantity of all radars to the l-th resource,
Figure GDA0003263385200000061
wherein, lambda is an adjusting factor influenced by the supply and demand difference.
The average price of the class i resources per task is
Figure GDA0003263385200000062
And step 3: solving system benefits
Step 3.1: determining task value
For the ith task demander diI-1, 2, …, M, the task needs to be completed before its task arrival time, only the response time TrLess than its cut-off time
Figure GDA0003263385200000063
Mine for mineCan execute its task; since the mission value decreases over time, the response times of different radars vary, and therefore the gains from different radar implementations also vary. The average value of the L resources of the ith task is
Figure GDA0003263385200000064
Step 3.2: determining resource consumption cost
For the j-th radar rjJ is 1,2, …, N, and the average cost per unit resource amount of L resources consumed when it executes the task, i.e., the cost, is determined according to the task start time and the supply price
Figure GDA0003263385200000065
Step 3.3: maximizing system revenue
The gain in completing the ith task is the product of its demand for all resources and the average value of the L resources, i.e.
Figure GDA0003263385200000066
Radar rjThe cost consumed is the product of the supply amount of the supply resource and the average cost per unit resource amount of the L kinds of resources, i.e. the cost
Figure GDA0003263385200000071
The system gain obtained by completing the ith task is
Figure GDA0003263385200000072
To maximize the system gain J in completing the ith taskiWill be selected
Figure GDA0003263385200000073
The radars are sorted according to the sequence of low resource consumption cost to high resource consumption cost, namely, the radars with low resource consumption are preferentially used for distributing tasks, and the sorted radar set is
Figure GDA0003263385200000074
And 4, step 4: task allocation
Step 4.1: determining task allocation order
And sequencing all task demanders according to the sequence of the task cut-off time from low to high, namely preferentially executing the task with the task cut-off time being earlier.
Step 4.2: determining whether a task can be successfully executed
Judgment of diThe required number x of L-th class resources, 1,2, …ilWith the selected set RiSum of the amount of resources that can be provided by the radar in (1)
Figure GDA0003263385200000075
If the supply and demand relationship is
Figure GDA0003263385200000076
Namely, the demand is larger than the supply quantity, the task failure is judged, and the number M of the task failure is calculatedfailAnd adding 1, and turning to the step 4.4, otherwise, turning to the step 4.5.
Step 4.3: constant task completion rate
Number of failures in task MfailIf the response time T of the radar is adjusted to be more than 0, namely the task completion rate is less than 100 percentrδ, go to step 4.2, reselect the selection response time TrRadars less than their cut-off time, i.e. update set Ri. The average cost of the L resources consumed becomes
Figure GDA0003263385200000077
Where δ is the response time change factor.
Step 4.4: task allocation process
Judgment of diThe required number x of L-th class resources, 1,2, …ilWhether or not greater than radar r1Number y that can be provided1lIf x isil≤y1lThen, indicate diTask of (1) is represented by1Finish, update r1Resource information of y1l-xilGo to the next class of resource, otherwise indicate r1Cannot satisfy d aloneiD is updated if the demand for the l-th resource is satisfiediIs xil-y1lAnd turn to r2；
Judgment of diAnd r and the remaining demand resources of2The size of the number of resources that can be provided, if (x)il-y1l)≤y2lThen, indicate diAnd (4) completing the distribution of the demands of the L-type resources of the L, L-1, 2 and …, and updating the radar r1And r2The resource information of (d) is allocated to the next kind of resource, otherwise, the resource information of (d) is updatediIs xil-y1l-y2lIs turned to r3(ii) a And so on until d is finishediFor the L, L-1, 2, …, L-type resource allocation, the next type of resource allocation is turned to. Repeat step 4.5 until d is completeiSuccessful allocation of all types of resources indicates diThe task of (2) is successfully allocated and the next task is shifted to. And repeating the steps 4.2-4.5 until the distribution of all tasks is completed.
The effect of the invention is further illustrated by the following simulation test:
simulation scene: suppose the system has 20 task requesters, 10 radars, and A, B, C types of resources. The required quantity of each type of resource of each task is subject to U (0,20) (U (a, b) represents that the value is uniform from a to b), the quantity of the resource which can be provided by each radar is subject to U (0,40), and the cut-off time of the task and the response time of the radar are subject to U (5, 10). Specific parameters of the demand information of the task demander and the supply information of the radar are shown in table 1 and table 2, respectively.
TABLE 1 task Requirements requirement information
Figure GDA0003263385200000081
Figure GDA0003263385200000091
TABLE 2 supply information for radar
Radar Resource A Resource B Resource C Response time Radar Resource A Resource B Resource C Response time
r
1 13 20 11 5.95 r 6 19 15 2 6.41
r 2 12 1 19 5.43 r 7 14 10 2 6.01
r 3 17 15 30 8.89 r 8 10 40 20 7.82
r4 29 20 16 9.16 r 9 20 14 33 8.38
r5 21 12 7 5.68 r 10 13 9 6 8.8
As can be seen from step 2, the resource price is determined by the amount of resource demand and the amount of radar supply for the task. The average price of the l-class resources for each task can be obtained according to formula (2), as shown in fig. 3.
All task requesters and radars are sorted in ascending order according to their respective task arrival times and response times, and based on the information provided in tables 1 and 2, we can know that the task execution order of the task requesters is { d }15，d3，d6，d9，d2，d5，d8，d13，d1，d4，d14，d10，d11，d7，d17，d20，d12，d12，d16，d19，d18The supply order of radar is { r }7，r6，r4，r8，r10，r9，r3，r5，r2，r1}。
First to d15The task of (1) is allocated, and the set of radars earlier than the task arrival time is R15＝{r7,r6,r4,r8,r10,r9,r3,r5,r2,r1}。d15It needs 4 units of resources A, 2 units of resources B, and 4 units of resources C to complete its task, and it can be seen from Table 2 that r714 units of resource a, 10 units of resource B, 2 units of resource C may be provided. Then resource a and resource B can both be satisfied and still require 2 units of resource C to complete their tasks, while the remaining resources are supplied with r second in the order6To r is provided, and r6Can meet the requirement of 2 units of resources C. To this point d15Is completed by r7Providing 4 units of resource A, 2 units of resource B, r7、r 62 units of resource C are provided. Update r7Is {10,8,0}, r6Is available with resources of 19,15,0, and continues to execute d ranked second in the execution of the task3The task of (2). And so on until all tasks are assigned. The specific distribution results are shown in table 3.
TABLE 3 task assignment results
Task demander Resource A Resource B Resource C
d1 r8,3 r8,10 r9,4
d2 r6,6 r4,16 r8,7；r10,1
d3 0 r7,2 r4,7
d4 r8,5 r8,7；r10,6 r3,6；r9,8
d5 r4,11 r8,14 r9,5；r10,5
d6 r7,8 r7,6；r6,3 r8,7；r4,9
d7 r3,14；r9,1 r4,9 r4,9
d8 r4,16 r8,6 r9,1
d9 r6,13；r7,2 r4,4；r6,12 r8,6
d10 r9,1 r9,9 r3,1
d11 r9,15 r3,3；r9,5 0
d12 r10,10 r1,4 r2,4
d13 r4,2；r8,2 r8,3 r9,5
d14 r9,3；r10,13 r10,3 r3,6
d15 r7,4 r7,2 r6,2；r7,2
d16 r1,3；r2,2 0 r2,1
d17 r3,3；r5,7 r3,3；r5,7 r3,3；r5,1
d18 r1,7 r1,2 r1,11；r2,2
d19 r1,3 r1,10 r2,6
d20 r5,14 r1,4；r2,1；r5,5 r2,6；r5,6
The completion rate of the tasks can be seen from fig. 4, in which the red triangle line shows the case of adopting a constant task completion rate, and the completion rate of the tasks is irrelevant to the number of the tasks, so that all the tasks are always successfully executed; the blue square line represents the case without constant task completion rate, and it can be seen that the task completion rate is worse and worse as the number of tasks increases.
FIG. 5 shows the system gains with and without constant task completion rates. It can be seen that the system benefit is higher when a constant task completion rate is employed than when no constant task completion rate is present due to the increased number of successfully completed tasks.
According to the specific implementation mode of the invention, the problem that the task value is time-varying in practical application and the task cannot be effectively distributed when the radar cannot process the task in real time can be well solved.

Claims (2)

1. A radar network task allocation method based on time efficiency constraint combination bilateral auction comprises the following steps:
step 1: collecting demand and supply information;
the auction center collects M task requesters D ═ D1,d2,…,dM]Requirement information of { quantity X of required resources, resource type L, task deadline TdR and N radars R ═ R1,r2,…,rN]Supply information { amount of supply resources Y, resource type L, response time Tr}；
Step 2: determining the price of the resource;
for each task, the auction center determines the price of the resource according to the supply-demand relationship of the radar resource, namely the price of the l-th resource of the ith task is determined by the difference value of the required quantity of the l-th resource of the ith task and the total supply quantity of all radars to the l-th resource;
and step 3: solving the system income;
total profit from completion of all tasks and consumption of radar-providing resourcesThe difference value of the costs is the total income of the system, the maximization of the income of the system is taken as a distribution criterion, and when the income of the system reaches the maximum, the task is indicated to be optimally distributed; for diI-1, 2, …, M, the task needs to be completed before its task arrival time, only the response time TrLess than its cut-off time
Figure FDA0003263385190000011
The radar can perform its task; as the task value is reduced along with the time, the response time of different radars is different, so the gains obtained by the execution of different radars are different; in order to maximize the system benefit when the ith task is completed, the selected radars are sequenced according to the sequence of the resource consumption cost from low to high, namely, the radars with low resource consumption are preferentially used for distributing the tasks, and the sequenced radars from high to low are collected into
Figure FDA0003263385190000012
The specific method comprises the following steps:
step 3.1: determining a task value;
for the ith task, i ═ 1,2, …, M, the response time T is selectedrLess than its cut-off time
Figure FDA0003263385190000013
The radar performs its task; average value of its L resources
Figure FDA0003263385190000014
Comprises the following steps:
Figure FDA0003263385190000015
wherein: t isi dIndicates the time-to-arrival of the ith task,
Figure FDA0003263385190000016
indicating the j-th radarResponse time, pilThe price of the l type resource of the ith task is represented;
step 3.2: determining a resource consumption cost;
for the j-th radar rjWhere j is 1,2, …, N, the average cost per unit resource amount of L resources consumed in executing a task, i.e., the cost, is determined based on the task start time and the supply price
Figure FDA0003263385190000021
Step 3.3: the total profit of the system is maximized;
the gain in completing the ith task is the product of its demand for all resources and the average value of the L resources, i.e.
Figure FDA0003263385190000022
Radar rjThe cost consumed is the product of the supply amount of the supply resource and the average cost per unit resource amount of the L kinds of resources, i.e. the cost
Figure FDA0003263385190000023
The system gain obtained by completing the ith task is
Figure FDA0003263385190000024
To maximize the system gain J in completing the ith taskiWill be selected
Figure FDA0003263385190000025
The radars are sorted according to the sequence of the resource consumption cost from low to high, namely, the radars with low resource consumption are preferentially used for distributing and sorting tasksThe latter radar set being
Figure FDA0003263385190000026
And 4, step 4: distributing tasks;
step 4.1: determining a task allocation sequence;
sequencing all task demanders according to the sequence of the task cut-off time from low to high, namely preferentially executing the task with the task cut-off time being earlier;
step 4.2: judging whether the task can be successfully executed;
determining task demander diNumber of demands on class I resource xilAnd the sum of the number of resources provided by the selected radar
Figure FDA0003263385190000027
Wherein L is 1,2, …, L; if it is
Figure FDA0003263385190000028
Namely, the demand is larger than the supply quantity, the task failure is judged, and the number M of the task failure is calculatedfailAdding 1, turning to step 4.4, otherwise, turning to step 4.5;
step 4.3: the task completion rate is constant;
number of failures in task MfailIf the response time T is more than 0, the response time T of the radar is adjustedr- δ, where δ denotes the adjustment step size determined according to the actual situation, increasing the chance of radar selection, go to step 4.2, reselect the selection response time TrRadar less than its arrival time;
step 4.4: a task allocation process;
determining task demander diNumber of demands on class I resource xilWhether or not greater than radar r1Number y that can be provided1lIf x isil≤y1lThen, indicate diTask of (1) is represented by1Finish, update radar r1Resource information of y1l-xilGo to the next class of resource, otherwise indicate r1Cannot be separatedSatisfy diD is updated if the demand for the l-th resource is satisfiediIs xil-y1lAnd steering radar r2；
Determining task demander diThe remaining demand resource of (2) and radar r2The size of the number of resources that can be provided, if (x)il-y1l)≤y2lThen, indicate diThe demand allocation of the l-type resource is completed, and the radar r is updated1And r2The resource information of (d) is allocated to the next kind of resource, otherwise, the resource information of (d) is updatediIs xil-y1l-y2lIs turned to r3(ii) a And so on until d is finishediThe distribution of the l type resource is turned to the distribution of the next type resource; repeat step 4.5 until d is completeiSuccessful allocation of all types of resources indicates diThe task is successfully distributed, and the next task is turned to; and repeating the steps 4.2-4.5 until the distribution of all tasks is completed.
2. The radar network task allocation method based on time-dependent constraint combination bilateral auction as claimed in claim 1, wherein the following formula is adopted in step 2 to determine the price p of the l-th resource of the i-th taskil；
Figure FDA0003263385190000031
Wherein, lambda is an adjusting factor influenced by the supply and demand difference quantity, and is determined according to the actual situation, and xilRepresenting the required quantity of the ith task to the l-th type resource, yjlIndicating the supply quantity of the jth radar to the l-th type resource.
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