A scheduling optimization method for the last train of urban rail transit under random passenger flow demand

Abstract

The present invention discloses a method for optimizing the scheduling of the last train of urban rail transit under random passenger flow demand, including: collecting basic information of the line network to be optimized, combining the line network OD demand, analyzing the change process of the transfer passenger flow demand during the last train operation period, and constructing a last train schedule optimization adjustment model with the minimum schedule adjustment cost and the maximum number of passengers as the optimization objectives; using a linearization method to convert nonlinear constraints into linear constraints; using a linear weighted method to convert a dual objective function into a single objective function, and using an optimization solver Cplex to solve the optimal solution of the optimization adjustment model to obtain the minimum cost of the last train schedule adjustment plan for urban rail transit. The present invention can further increase the number of passengers who can transfer while ensuring that the line network pays the minimum cost, and has more significant advantages in saving operating costs and improving passenger service levels.

Description

Technical Field

[0001] The present invention relates to the technical field of urban rail transit operation organization scheme optimization, and more specifically, to an urban rail transit last train schedule adjustment scheme considering minimum cost under random passenger flow demand. Background Art

[0002] With the rapid development of urban rail transit networks, more and more passengers travel by urban rail transit late at night. However, for maintenance and cost-saving considerations, urban rail transit rarely provides 24-hour service. In this case, passengers are forced to use other modes of transportation, such as taxis, night buses, etc. to reach their destinations. This problem is particularly prominent in large cities with larger subway networks. The lack of effective connection between the last trains of each line poses a challenge to both subway operators and passengers. At the same time, subway operators face problems such as the inability to effectively carry passengers on some lines, resulting in inefficient use of transportation resources. On the other hand, due to the lack of effective connection, passengers will have to seek other modes of transportation, which will incur more time and economic costs.

[0003] The last train scheduling problem is a complex task due to the diversity of urban rail transit network structures and the uncertainty of passenger demand during the last train. Although the existing AFC system can record the information of passenger arrival and departure, there is still a lot of uncertainty in accurately estimating the passenger demand during the last train. The randomness of passenger demand is the primary challenge in solving the train scheduling problem. In addition, adjusting the departure time of the last train will seriously disrupt the original train operation plan, bringing additional costs to both the operator and passengers. The more departure times that need to be adjusted, the greater the interference with the original train schedule, which leads to increased travel costs for passengers. Managing additional adjustment costs becomes the second challenge in dealing with the last train scheduling problem. Therefore, designing a coordinated last train schedule considering the uncertainty of passenger demand and the additional cost of adjusting the last train can increase passenger accessibility, reduce travel time, and ultimately improve the overall service quality. Summary of the invention

[0004] In order to solve the above technical problems, the present invention proposes a scheduling optimization method for the last train of urban rail transit under random passenger flow demand, and constructs a mixed integer programming model to optimize the last train schedule of each line. While reducing the inaccessibility of passengers, it further takes into account the randomness of late-night passenger demand and the related adjustment costs of subway operators.

[0005] The present invention provides a method for optimizing the scheduling of the last train of urban rail transit under random passenger flow demand, comprising:

[0006] S1: Collect basic information of the line network to be optimized, including the existing timetable, the running time of each section of the line network, the stop time of each station, the OD demand of the line network during the optimization period, and the minimum transfer time of each transfer path;

[0007] S2: Combined with the OD demand of the line network, the change of transfer passenger flow demand during the last train operation period is analyzed, and a last train schedule optimization adjustment model is constructed with the minimum schedule adjustment cost and the maximum achievable passenger flow as the optimization objectives;

[0008] S3: In the optimization adjustment model, linearization method is used to transform nonlinear constraints into linear constraints;

[0009] S4: using a linear weighted method to transform the dual objective function into a single objective function, and using the optimization solver Cplex to solve the optimal solution of the optimization adjustment model, to obtain the minimum cost urban rail transit last train schedule adjustment plan;

[0010] S5: Evaluate the adjustment plan for the last train schedule of the urban rail transit through preset indicators.

[0011] In this scheme, a last train schedule optimization adjustment model is constructed with the optimization objectives of minimizing the schedule adjustment cost and maximizing the achievable passenger flow. The schedule adjustment cost is specifically:

[0012] Analyze the timetable adjustment cost based on the number of lines affected by the change in arrival and departure times, the number of stations affected by the adjustment in arrival and departure times, and the total amount of time that the arrival and departure times of each affected station need to be changed;

[0013] Introducing 0-1 variables Determine whether the arrival and departure time of the last train at the station has been adjusted, and introduce variables To indicate the line Whether adjustment occurs, the adjustment cost objective function is divided into the sub-objective function of adjusting the number of lines , adjust the station number sub-objective function And the total adjustment time sub-objective function , respectively expressed as: in, , Trains on the line The station on The original departure time of , Trains on the line Get on the train station The departure time, is a collection of lines, For line Meet at the station on the way.

[0014] In this scheme, a last train schedule optimization adjustment model is constructed with the optimization objectives of minimizing the schedule adjustment cost and maximizing the achievable passenger flow. The achievable passenger flow is specifically:

[0015] Introducing 0-1 variables Describe the transfer route Accessibility, transfer paths Spare time for transfers With line and At the station Arrival and departure time , The relationship between them is expressed as: in For transfer path The OD passenger flow between stations is converted into the transfer passenger flow on each transfer path. Initial transfer passenger demand , and quantify the randomness through the impact of the accessibility of the passenger's previous transfer path, and obtain the random passenger flow demand, which is expressed as: in, For transfer path The initial transfer passenger demand on is the ratio of unreachable transfer directions to the total preceding transfer directions, is the sensitivity parameter, is the passenger demand reduction rate; the achievable passenger flow objective function for: in is a collection of lines, For line Meet at the station on line Meet at the station on the way.

[0016] In this solution, the optimization adjustment model is subject to interval time constraints, stop time constraints, early and late departure time constraints, priority connection constraints and transfer accessibility constraints.

[0017] In this scheme, a linearization method is used in the optimization adjustment model to transform nonlinear constraints into linear constraints, specifically:

[0018] The transfer accessibility constraint is linearized and expressed as: ;

[0019] The adjustment cost is linearized and expressed as: ;

[0020] Demand for random passenger flow Linearize and introduce variables Expression Line Train at the station The success rate of the previous transfer direction is , random passenger flow demand is expressed as: ;in, is a positive infinite quantity, The proportion of transfer directions that are unreachable in the previous order The corresponding passenger flow reduction coefficient is It is a 0-1 variable describing the transfer path accessibility, For transfer path The extra time for transfers, It is a 0-1 variable describing whether to connect the line Make adjustments, It is a 0-1 variable describing the train on the line. The station on Whether the arrival and departure times are adjusted, For transfer path The initial transfer passenger flow demand, , For line ,line Meet at the station on For station indicators.

[0021] In this solution, the adjustment cost objective function and the achievable passenger flow objective function are converted into the total cost in the optimization adjustment model, and the double objective function is converted into a single objective function using a linear weighted method, which is expressed as: in, , , , is the adjustment factor, , , , is the weight coefficient, , , They are the sub-objective function for adjusting the number of lines, the sub-objective function for adjusting the number of stations, and the sub-objective function for adjusting the total adjustment time in the adjustment cost objective function. is the objective function of achievable passenger flow;

[0022] The optimization solver Cplex is used to solve the optimal solution of the optimization adjustment model and generate the minimum cost adjustment plan for the last train schedule of urban rail transit.

[0023] In this scheme, the original timetable of the last train of urban rail transit on weekdays and special weekdays is obtained, and the total accessible transfer passenger flow, the number of adjusted stations, the number of adjusted lines, and the total adjusted time are used as preset indicators;

[0024] The original timetable is compared with the adjustment plan for the last train timetable of urban rail transit to obtain index parameters, and the advantages, disadvantages and applicability of the adjustment plan for the last train timetable of urban rail transit are obtained by presetting index thresholds and index parameters.

[0025] Compared with the prior art, the present invention has the following beneficial effects:

[0026] The present invention models and optimizes the urban rail transit timetable from the perspective of "minimum cost under random passenger flow demand", thereby reducing the inaccessibility of passengers, reducing travel time, and ultimately improving the overall service quality;

[0027] Although the last train timetable adjustment plan for urban rail transit that considers the minimum cost under random passenger flow demand limits the increase in the number of passengers that can be reached, it can greatly reduce the number of adjusted lines, stations and total adjustment time, and significantly reduce the additional cost paid by rail transit operators for adjusting the timetable. BRIEF DESCRIPTION OF THE DRAWINGS

[0028] In order to more clearly illustrate the technical solutions in the embodiments or exemplary embodiments of the present invention, the drawings required for use in the embodiments or exemplary descriptions will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present invention. For ordinary technicians in this field, other drawings can be obtained according to the drawings without paying creative work.

[0029] Figure 1 A flow chart of a method for optimizing the scheduling of the last train of urban rail transit under random passenger flow demand provided by an embodiment of the present invention is shown;

[0030] Figure 2 A schematic diagram of passenger transfer behavior during the last train operation period provided by an embodiment of the present invention is shown;

[0031] Figure 3 A schematic diagram showing the composition of random passenger flow demand during the last train departure period provided by an embodiment of the present invention is shown;

[0032] Figure 4 A schematic diagram of OD passenger flow path allocation provided by an embodiment of the present invention is shown;

[0033] Figure 5 The different sensitivity parameters provided by the embodiments of the present invention are shown Schematic diagram of the impact of different values ​​on the demand reduction rate. DETAILED DESCRIPTION

[0034] In order to more clearly understand the above-mentioned purpose, features and advantages of the present invention, the present invention is further described in detail below in conjunction with the accompanying drawings and specific embodiments. It should be noted that the embodiments of the present application and the features in the embodiments can be combined with each other without conflict.

[0035] In the following description, many specific details are set forth to facilitate a full understanding of the present invention. However, the present invention may also be implemented in other ways different from those described herein. Therefore, the protection scope of the present invention is not limited to the specific embodiments disclosed below.

[0036] Figure 1 A flow chart of a method for optimizing the scheduling of the last train of urban rail transit under random passenger flow demand provided by an embodiment of the present invention is shown.

[0037] The embodiment of the present invention provides a method for optimizing the scheduling of the last train of urban rail transit under random passenger flow demand, comprising:

[0038] S1: Collect basic information of the line network to be optimized, including the existing timetable, the running time of each section of the line network, the stop time of each station, the OD demand of the line network during the optimization period, and the minimum transfer time of each transfer path;

[0039] S2: Combined with the OD demand of the line network, the change of transfer passenger flow demand during the last train operation period is analyzed, and a last train schedule optimization adjustment model is constructed with the minimum schedule adjustment cost and the maximum achievable passenger flow as the optimization objectives;

[0040] In the embodiment of the present invention, when constructing the model, the following assumptions are made: (1) The up and down directions of each line are regarded as two independent lines and do not interfere with each other; (2) The minimum transfer time between each transfer path in the transfer station is known and fixed; (3) The trains on each line run strictly according to the generated timetable, without skipping stops, and do not consider abnormal operation such as delays; (4) Whether the train passenger load exceeds the capacity limit is not considered. An optimization adjustment model is established based on the optimization principle of the last train timetable of urban rail transit. The optimization result must meet the constraints of interval operation time, stop time, morning and evening departure time, priority connection direction, and accessibility of transfer paths, and be compatible with actual passenger flow demand. ‌OD demand refers to the traffic volume, that is, the traffic volume at the starting point and end point of the trip. The optimization adjustment model has two goals. The first is to minimize the cost of timetable adjustment, which includes the number of adjusted lines, the number of adjusted stations, and the total adjustment time; the second is to maximize the transfer passenger flow.

[0041] In order to improve the connectivity of the last train, the subway operator needs to adjust the arrival and departure times of the last train at specific stations on specific lines. The more stations and last train departure times are adjusted, the higher the related costs. The cost of schedule adjustment is analyzed based on the number of lines affected by the change in arrival and departure times, the number of stations affected by the adjustment of arrival and departure times, and the total amount of time that the arrival and departure times of each affected station need to be changed; introduce 0-1 variables Determine whether the arrival and departure time of the last train at the station has been adjusted, and introduce variables To indicate the line Whether adjustments have occurred, The expression is: The expression is: ,like This indicates that the line The station on The last train timetable has been adjusted, otherwise it has not been adjusted. Similarly, the cost adjustment objective function is divided into the line number adjustment sub-objective function , adjust the station number sub-objective function And the total adjustment time sub-objective function , respectively expressed as: in, , Trains on the line The station on The original departure time of , Trains on the line Get on the train station The departure time, is a collection of lines, For line Meet at the station on the way.

[0042] According to an embodiment of the present invention, the transfer accessible passenger flow is maximized in the optimization adjustment model. Figure 2 Showing the line and At the station When the intersection occurs, the passenger transfer is accessible or inaccessible, and the transfer time composition. Introducing 0-1 variables Describe the transfer route Accessibility, transfer paths Spare time for transfers With line and At the station Arrival and departure time , The relationship between them is expressed as: in For transfer path Transfer time on

[0043] like Figure 3 As shown in the figure, the OD passenger flow between stations is converted into the transfer passenger flow on each transfer path. Initial transfer passenger demand , expressed as: in For transfer path Is it on the OD path? If yes, it is 1, otherwise it is 0; OD pair for the last train operation period The initial passenger flow demand, For OD Select Path probability.

[0044] At the same time, previous transfer accessibility analysis shows that within the subway network, there is a certain degree of randomness in the passengers entering and leaving the transfer station, which is affected by the accessibility of the previous transfer path. Figure 4 The specific relationship between transfer accessibility and random passenger demand is shown. The randomness is quantified by the impact of the accessibility of the passenger's previous transfer path, and the random passenger flow demand is obtained, which is expressed as: in, For transfer path The initial transfer passenger demand on is the ratio of unreachable transfer directions to the total preceding transfer directions, is the sensitivity parameter, is the passenger demand reduction rate, different The impact of the value on the demand reduction rate is as follows Figure 5 As shown, is the line indicator, Gather for the station, For station indicators;

[0045] Accessible passenger flow objective function for: in is a collection of lines, For line Meet at the station on line Meet at the station on the way.

[0046] S3: In the optimization adjustment model, linearization method is used to transform nonlinear constraints into linear constraints;

[0047] The optimization adjustment model is subject to interval time constraints, stop time constraints, early and late departure time constraints, priority connection constraints and transfer accessibility constraints;

[0048] Interval time constraint, introducing interval operation time adjustment coefficient To describe the train from the station arrive The running time changes and restrictions , expressed as: ;in For trains on the line From the station arrive The running time of

[0049] Stop time constraints, introduce stop time adjustment coefficient To describe the line Train at the station The change of stop time and limit ,express: ;

[0050] To ensure a relatively stable time arrangement when trains depart from the departure station, the departure time of the last train of each line at its first station is constrained, which can be expressed as: , The earliest departure time is The latest departure time;

[0051] Priority connection constraint, in order to maintain a high passenger service quality during the operation of the last train, subway operators often define certain priority transfer directions as "priority connection directions". In order to ensure that the last train can reach these key transfer paths, it is expressed as: ; is a set of strongly connected transfer paths; transfer accessibility constraints, introducing 0-1 variables To describe the transfer direction If the accessibility Passengers can transfer in the direction On the line Switch to line ,otherwise .

[0052] The transfer accessibility constraint is linearized and expressed as , is a positive infinite quantity;

[0053] The adjustment cost is linearized and expressed as: ; Demand for random passenger flow Linearize and introduce variables Expression Line Train at the station The success rate of the previous transfer direction is , expressed as: After linearization, the random passenger flow demand is expressed as: ;in, The proportion of transfer directions that are unreachable in the previous order The corresponding passenger flow reduction coefficient is expressed linearly, and the passenger flow objective function can be expressed as For nonlinear elements, auxiliary variables are introduced to linearize them. To represent the nonlinear element, it is expressed as The accessible passenger flow objective function can be rewritten as: .

[0054] S4: using a linear weighted method to transform the dual objective function into a single objective function, and using the optimization solver Cplex to solve the optimal solution of the optimization adjustment model, to obtain the minimum cost urban rail transit last train schedule adjustment plan;

[0055] In the optimization adjustment model, the adjustment cost objective function and the achievable passenger flow objective function are converted into the total cost, and the double objective function is converted into a single objective function using a linear weighted method, which is expressed as: in, , , , is the adjustment factor, , , , is the weight coefficient, , , They are the sub-objective function for adjusting the number of lines, the sub-objective function for adjusting the number of stations, and the sub-objective function for adjusting the total adjustment time in the adjustment cost objective function. is the objective function of achievable passenger flow;

[0056] The optimization solver Cplex is used to solve the optimal solution of the optimization adjustment model and generate the minimum cost adjustment plan for the last train schedule of urban rail transit.

[0057] S5: Evaluate the adjustment plan for the last train schedule of the urban rail transit through preset indicators.

[0058] The original timetable of the last train of urban rail transit on weekdays and special weekdays is obtained, and the total achievable transfer passenger flow, the number of adjusted stations, the number of adjusted lines, and the total adjusted time are used as preset indicators. The indicator that reflects the passenger service quality is: the total achievable transfer passenger flow; the indicators that reflect the additional expenditure cost are: the number of adjusted stations, the number of adjusted lines, and the total adjusted time.

[0059] The original timetable is compared with the adjustment plan for the last train timetable of urban rail transit to obtain index parameters, and the advantages, disadvantages and applicability of the adjustment plan for the last train timetable of urban rail transit are obtained by presetting index thresholds and index parameters.

[0060] In the several embodiments provided in the present application, it should be understood that the disclosed devices and methods can be implemented in other ways. The device embodiments described above are only schematic. For example, the division of the units is only a logical function division. There may be other division methods in actual implementation, such as: multiple units or components can be combined, or can be integrated into another system, or some features can be ignored or not executed. In addition, the coupling, direct coupling, or communication connection between the components shown or discussed can be through some interfaces, and the indirect coupling or communication connection of the devices or units can be electrical, mechanical or other forms.

[0061] The units described above as separate components may or may not be physically separated, and the components displayed as units may or may not be physical units; they may be located in one place or distributed on multiple network units; some or all of the units may be selected according to actual needs to achieve the purpose of the present embodiment.

[0062] In addition, all functional units in the embodiments of the present invention may be integrated into one processing unit, or each unit may be separately used as a unit, or two or more units may be integrated into one unit; the above-mentioned integrated units may be implemented in the form of hardware or in the form of hardware plus software functional units.

[0063] Those skilled in the art can understand that all or part of the steps of implementing the above method embodiments can be completed by hardware related to program instructions, and the aforementioned program can be stored in a computer-readable storage medium. When the program is executed, it executes the steps of the above method embodiments; and the aforementioned storage medium includes: mobile storage devices, read-only memories (ROM, Read-Only Memory), random access memories (RAM, Random Access Memory), disks or optical disks, and other media that can store program codes.

[0064] Alternatively, if the above-mentioned integrated unit of the present invention is implemented in the form of a software function module and sold or used as an independent product, it can also be stored in a computer-readable storage medium. Based on this understanding, the technical solution of the embodiment of the present invention can be essentially or partly reflected in the form of a software product that contributes to the prior art. The computer software product is stored in a storage medium and includes several instructions for a computer device (which can be a personal computer, server, or network device, etc.) to execute all or part of the methods described in each embodiment of the present invention. The aforementioned storage medium includes: various media that can store program codes, such as mobile storage devices, ROM, RAM, magnetic disks or optical disks.

[0065] The above is only a specific embodiment of the present invention, but the protection scope of the present invention is not limited thereto. Any person skilled in the art can easily think of changes or substitutions within the technical scope disclosed by the present invention, which should be included in the protection scope of the present invention. Therefore, the protection scope of the present invention should be based on the protection scope of the claims.

Claims

1. A method for optimizing the scheduling of the last train of urban rail transit under random passenger flow demand, characterized in that: The following steps are involved: S1: Collect basic information of the line network to be optimized, including the existing timetable, the running time of each section of the line network, the stop time of each station, the OD demand of the line network during the optimization period, and the minimum transfer time of each transfer path; S2: Combined with the OD demand of the line network, the change of transfer passenger flow demand during the last train operation period is analyzed, and a last train schedule optimization adjustment model is constructed with the minimum schedule adjustment cost and the maximum achievable passenger flow as the optimization objectives; S3: In the optimization adjustment model, linearization method is used to transform nonlinear constraints into linear constraints; S4: using a linear weighted method to transform the dual objective function into a single objective function, and using the optimization solver Cplex to solve the optimal solution of the optimization adjustment model, to obtain the minimum cost urban rail transit last train schedule adjustment plan; S5: Evaluate the adjustment plan of the last train schedule of the urban rail transit by using preset indicators; A last train schedule optimization adjustment model is constructed with the optimization objectives of minimizing the schedule adjustment cost and maximizing the achievable passenger flow. The schedule adjustment cost is specifically: Analyze the timetable adjustment cost based on the number of lines affected by the change in arrival and departure times, the number of stations affected by the adjustment in arrival and departure times, and the total amount of time that the arrival and departure times of each affected station need to be changed; Introducing 0-1 variables Determine whether the arrival and departure time of the last train at the station has been adjusted, and introduce variables To indicate the line Whether adjustment occurs, the adjustment cost objective function is divided into the sub-objective function of adjusting the number of lines , adjust the station number sub-objective function And the total adjustment time sub-objective function , respectively expressed as: in, , Trains on the line The station on The original departure time of , Trains on the line Get on the train station The departure time, is a collection of lines, For Line The station set on the last train schedule is constructed to minimize the schedule adjustment cost and maximize the achievable passenger flow as the optimization objectives, and the achievable passenger flow is specifically: Introducing 0-1 variables Describe the transfer route Accessibility, transfer paths Spare time for transfers With line and At the station , Arrival and departure time , The relationship between them is expressed as: in For transfer path The OD passenger flow between stations is converted into the transfer passenger flow on each transfer path. Initial transfer passenger demand , and quantify the randomness through the impact of the accessibility of the passenger's previous transfer path, and obtain the random passenger flow demand, which is expressed as: in, For transfer path The initial transfer passenger demand on The proportion of unreachable transfer directions to the total number of preceding transfer directions, is the sensitivity parameter, is the passenger demand reduction rate; Accessible passenger flow objective function for: in is a collection of lines, For Line Meet at the station on For Line Meet at the station on the way.

2. The method for optimizing the last train scheduling of urban rail transit under random passenger flow demand according to claim 1, characterized in that: The optimization adjustment model is subject to interval time constraints, stop time constraints, early and late departure time constraints, priority connection constraints and transfer accessibility constraints.

3. The method for optimizing the scheduling of the last train of urban rail transit under random passenger flow demand according to claim 1 is characterized in that: In the optimization adjustment model, the linearization method is used to transform the nonlinear constraints into linear constraints, specifically: The transfer accessibility constraint is linearized and expressed as: ; The adjustment cost is linearized and expressed as: ; Demand for random passenger flow Linearize and introduce variables Expression Line Train at the station The success rate of the previous transfer direction is , random passenger flow demand is expressed as: ;in, is a positive infinite quantity, The proportion of transfer directions that are unreachable in the previous order The corresponding passenger flow reduction coefficient is It is a 0-1 variable describing the transfer path accessibility, For transfer path The extra time for transfers, , Trains on the line The station on Arrival and departure times, , For trains on the line The station on The original arrival and departure time, It is a 0-1 variable describing whether to connect the line Make adjustments, It is a 0-1 variable describing the train on the line. The station on Whether the arrival and departure times are adjusted, For transfer path The initial transfer passenger flow demand, , For Line ,line Meet at the station on For station indicators.

4. The method for optimizing the last train scheduling of urban rail transit under random passenger flow demand according to claim 1, characterized in that: In the optimization adjustment model, the adjustment cost objective function and the achievable passenger flow objective function are converted into the total cost, and the double objective function is converted into a single objective function using a linear weighted method, which is expressed as: in, is the adjustment factor, is the weight coefficient, They are the sub-objective function for adjusting the number of lines, the sub-objective function for adjusting the number of stations, and the sub-objective function for adjusting the total adjustment time in the adjustment cost objective function. is the achievable passenger flow objective function; the optimization solver Cplex is used to solve the optimal solution of the optimization adjustment model to generate the minimum cost urban rail transit last train schedule adjustment plan.

5. The method for optimizing the scheduling of the last train of urban rail transit under random passenger flow demand according to claim 1 is characterized in that: Obtain the original timetable of the last train of urban rail transit on weekdays and special weekdays, and use the total accessible transfer passenger flow, the number of adjusted stations, the number of adjusted lines, and the total adjusted time as preset indicators; The original timetable is compared with the adjustment plan for the last train timetable of urban rail transit to obtain index parameters, and the advantages, disadvantages and applicability of the adjustment plan for the last train timetable of urban rail transit are obtained by presetting index thresholds and index parameters.

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