Best’11 strategy in cricket using MCDM,rough matrix and assignment model

Abstract

In 1982, Pawlak set up a fresh approach to deal with uncertainties namely rough set theory, Multiple-Criteria Decision Making (MCDM) first traced by Benjamin Franklin in 17th century. Several researchers did significant contribution to MCDM thereafter. An assignment problem involves what happens to the effective function when each of a number of sources is associated with the same number of destinations. Using MCDM, Rough matrices and Assignment model we are inducing an idea to pick Best’11 in all three formats (Test, One Day Internationals (ODI), Twenty20 International matches (T20I)) in the game of cricket with players from two nationals. Using the existing data, we are providing best batting position for any player to maximize team’s run. In addition, based on the preprocessing of informations, we are bringing some new indices to pick Indian squad for the 2019 World Cup cricket held in England from May 2019 to July 2019. After making a selection from our framework, we will compare the list of selected players by Board of Cricket Control Board in India (BCCI) and giveaway the percentage of similarity between the our selection against BCCI’s selection. We pick 11 players after selecting 15 players from 24 players to formulate the assignment model and offer the best batting order to optimize team’s run.

Keywords

Rough set rough matrix information systems MCDM best’11 assignment problem

1 Introduction

Pawlak [1] coined a prominent conception to deal with uncertainties namely rough sets with two approximations from the universe set. Many theories to deal with uncertainties exists. To name few, probability, fuzzy sets [5], vague set and interval mathematics. Rough membership function is stated by Pawlak [2, 8]. The ornamental of the rough set theory is that lack of clarity is personified by set boundary region rather than membership function. In 2013, Vijayabalaji and Balaji [6] gave a 3-valued matrix representation of rough sets whose elements are r_m(x)=1, if x belongs to lower approximation

r_m (x) =0, if x belongs to upper approximation

r_m (x) = (0, 1), if x belongs to boundary region and termed it as a rough matrix. In 2015,Vijayabalaji and Balaji [7] applied rough matrix theory in MCDM problems.The papers [12 –14] and [16] describes about the fusion of cricket game with many theories using different techniques.

Information systems (IS) [9] being a pedagogical frontier has its roots in 1960. He made a survey by prioritizing the counterplan of data processing applications in 10 organizations. An IS is 4-fold [8] IS = (U, Q, V_D, f) where U ≠ 0 is a set of finite objects; Q ≠ 0 is a finite set of characteristics; V_D =⋃ domains, V = ⋃ _q∈QV_q, where V_qis the value set of attribute q, called the domain of q; f : U * Q → V_D is an information function which assigns particular values from domains of attributes to objects such as ∀q ∈ Q, x ∈ U. f (q, x) ∈ V_q where f (q, x) denotes the value of attribute a for object x.

An assignment model‘s [10] objective is to find the change in the effectiveness function when each number of origin is associated with each number of destination in such a way as to extremize the overall effectiveness.A specific type of problem with linear programming is known as problem of assignment [10]. It deals with allocating various resources on a one-to-one basis to specific activities in such a way as to maximize the resulting effectiveness. The Mathematical representation of the assignment model [10] with the assignment of i^th member to j^th job (denoted by x_ij) is defined as,

$x_{ij} = {\begin{matrix} 0, & if the i^{th} member is not assigned to j^{th} job \\ 1, & if the i^{th} member is assigned to j^{th} job \end{matrix}$ i.e.,

Minimize $Z = \sum_{j = 1}^{n} \sum_{i = 1}^{n} c_{ij} x_{ij}$ (or) $Z = \sum_{i = 1}^{n} \sum_{j = 1}^{n} c_{ij} x_{ij}$

Subject to the constraints

$\sum_{j = 1}^{n} x_{ij} = 1, \begin{matrix} j = 1, 2, . . . \end{matrix}, n$ (one job is assigned to i^th member)

$\sum_{i = 1}^{n} x_{ij} = 1, \begin{matrix} j = 1, 2, . . . \end{matrix}, n$ (one member is assigned to the j^th job)

and $x_{ij} = \begin{matrix} 0 & or \end{matrix} \begin{matrix} 1 . \end{matrix}$

We shall have a Brief introduction about the game of cricket. Cricket is a game, played in a ground, with a rectangular pitch being in the centre of the play area called field. Three sticks with bails called as stumps are placed on either sides of the pitch. It’s a team game. It needs two teams to play cricket. Each team needs a minimum of 11 players. Say, if two teams, Team A and Team B are playing against each other. The game happens in two sessions. The team that will bat first will be decided by a toss. Say, When team A wins the toss and opts for batting, Then,in first session of play, two players from team A, will use a bat to hit the ball bowled by a bowler from team B. a wicketkeeper stands behind the stump. All the other 9 players from team B will try to stop the ball hit by the batsman, which is called as fielding. the batting team(team A) needs to hit most number of runs and set a target within a stipulated number of overs.in the second session of play, the target set by team A should be surpassed by team B. In the second session Team B will bat and Team A will field and bowl to defend the target they set in the first session. The game will be refereed by two umpires.one would stand behind the stumps in the non-striker end and the other would stand in the field at a position called square leg who is called “leg umpire”. The decision given the umpire is final. There is a third umpire who sits outside the field and gives decision if asked to review, by looking in the camera feed action replays. If the target set by team A is not surpassed by team B, then Team A wins the match. If the target set by team A is surpassed by team B by at least one run, the game is won by team B.If team A target are not met and the same scores are achieved, then they are called ’tie.’ The game is played in three formats, namely Test cricket-Played for five days or 2 innings, One Day International (ODI) (50 overs per team per session), Twenty 20 (T20) (A maximum of 20 overs or until the loss of 10 wickets whichever is earlier).

Test cricket is played for 5 days. Maximum of 90 overs per day can be bowled by the bowling team. Two innings can be played.in each innings both the team will bat and bowl once. If two teams play, say Team A and Team B.If A wins the toss and opts first for batting,, after team A’s batting is over, Team B will bat. This one cycle is called as one innings. After the first innings is over, second innings will start, Team A will bat in the second innings again followed by Team B. If Team B surpasses the runs scored by Team A in both innings by at least one run, then Team B is the winner. If Team B does not exceed the runs scored by Team A in both innings and loses all the wickets before score is reached, then Team A is the winner. If team B does not overtaking the score of team A in both the innings, but plays all through the rest of the days without losing all the wickets, then the result of the match is called ‘Draw’. In test cricket, there is no specific stipulation for a team to bat. If the first batting team bats for all the five days without losing all the wickets, it is allowed. In the first innings, the first batting team will score runs as much in as many days. The second batting team should score runs not less than half of the runs scored by the first batting team. If the second batting team fails to do so it will be forced to bat again in its second innings, this is called follow-on. If team A plays first sets a total, that the team B could not achieve even half the total runs, then Team would be forced to follow on. Even in the second innings if team B loses the all wickets before scores the runs scored by team A in the first innings. Then it is called as ‘Innings defeat’ by team A.

One Day International (ODI) is played by two teams for a day. The toss winning team will determine first whether to bat or bowl. If the toss winning team chooses to bat first, the first batting team will bat for 50 overs in that session and set a target. The team that bat later should surpass the target set by the first batting team by at least one run to win the match. When the first batting team is in a position to prevent the second batting team from achieving the set target of runs, the former is the winner. Unlike test cricket, the first batting team has to complete their batting in their first session within 50 overs. The bowling team should complete their 50 overs before the stipulated time else would face penalty for slow over rate.

Twenty 20 (T20) is similar to one day international match. But played by two teams for twenty overs apiece.in this format certain rules are exclusive to it, such as a no ball bowled by the bowler will invite a ‘free hit’, where in the next ball bowled by the bowler is awarded as free hit to the batting team. If the batsman is caught or bowled or stumped, it won’t be called as out.

We shall have a glance about frequently used batting terminologies. During batting by a team, two batsmen from the team will enter the ground; the rest will be non-playing outside the ground. The two batsmen will stand in either ends of the pitch. The batsmen who will face the ball from the bowler is called striker. The one who stands in the opposite end of the striker is called as non-striker. From the non-striker’s end, the bowler must bowl the ball. The batsman will face the ball bowled by the bowler, the batsmen will attempt to score as follows.

(1) Run: one running between the wickets is called as one run by hitting the ball with the bat, then running between the wickets until the ball is thrown back to meet the stump.

(2) Four: when a batsman hits the ball along the ground in a way, the fielders can’t stop the ball and pitches within the boundary line before touching the boundary line or crossing it, then is declared as four and four runs would be awarded to the batsman.

(3) Six: when a batsman hits the ball and the ball crosses the boundary line before hitting the ground, then is declared as ‘six’. Six runs would be awarded to the batsman.

We shall have a look about frequently used bowling terminologies.The bowler intention will be on two things, to get the batsmen out by the allowed methods of dismissals and also to restrict the batsman from scoring runs. The dismissal of a batsman can be happen in any one of the following predominant ways, the ball caught by the fielder after being hit by the batsman but before it touches the ground. The ball bowled by the bowler hits the stumps and dislodges the bails is known as bowled. When the batsmen run between the wickets and fail to successfully reach the crease line before the player of the fielding team dislodges the bails of the wicket at the respective end is known as run out. When the ball is bowled by the bowler and the ball hits the batsman’s leg below the knee before touching the bat in straight line with the stumps of the wicket is adjudged out using the terminology leg before wicket (LBW).

Glossary of terms in cricket are as follows.

Ground: The play field with some grass.

Pitch: It is the 22 yard hard surface on which the ball is bowled, the area between the wickets on either ends.

Wicket: the three stumps with a set of bails on the top of the stumps.

Umpire: The official authority present on the field to officiate the game.

Third Umpire: The official authority present outside the field to assist the on field umpires.

Batsman: The two players who play with the bat on the pitch.

Bowler: The bowler who bowls to the batsman.

Fielder: The player of the bowling team.

Wicket keeper: The player who stands behind the stumps and keeps wickets.

Over: Contains 6 legal deliveries which a bowler bowls at one go.

Innings: One innings played by a team this is in any one of the above mentioned three formats.

List of abbreviations used

GLB-greatest lower bound, Mat -matches played, Inns - innings (batted / bowled), NO-not out, HS-highest score,

Ave - average (bowler/batsman), BF- balls faced, SR-strike rate (bowler/batsman), I by C - index by cases, BatI1 - batting index 1, BatI2- batting index 2, OBatI - overall batting index, Econ-economy rate, W/I - wickets per innings, Top - batsman batted 1-3,Middle - batsman batted 4-7, Tail - batsman batted 8-11, I by D- bowlers index by dismissal information, BI - bowler’s index, OBowI- overall bowling index,Pacebat - fast cum handy batsman, Batpace - batsman cum handy pace bowler,SpinBat - spin cum handy batsman, BatSpin - batsman cum handy spin bowler, BatWK- batsman cum Wicket Keeper, O_Ave -overall average, O_SR- overall strike rate, O₁₀₀ - overall hundreds, O₅₀- overall fifties, ICO_Bat - overall index by cases dominant by bat, ICF_Bat - 2015-2019 cases by index dominant by bat, ICO_bow - overall index by cases dominant by bowl, ICF_Bow -2015-2019 index by cases dominant by bowl, BowI - bowling index, F_Ave- average between 2015-2019, F_SR - strike rate between 2015-2019,F₁₀₀ - hundreds between 2015-2019, F₅₀ -fifties between 2015-2019, DI_Bow -dismissal Index for bowler,O_W/I - overall wickets per innings. F_W/I -wickets per innings between 20015-2019,BowI - bowling index IVR- index by venues and results, IVR_H-IVR index for home venue, IVR_A-IVR index for away venue, IVR_N- IVR index for neutral venue, IVR_W- IVR index for won matches, IVR_L-IVR index for lost matches, ICV_bow - bowling index by cases using venues, ICR_bow - bowling index by cases using result of the match, IC_F - index by cases during 2015-2019, Ave_H -average at home, Ave_A- average at away, Ave_N- average at neutral, Ave_W - average in won matches, Ave_L - average in lost matches,SR_H-strike rate at home,SR_A-strike rate at away,SR_N-strike rate at neutral,SR_W-strike rate in won matches,SR_L-strike rate in lost matches, WI_H- wickets per innings at home,WI_A- wickets per innings at away,WI_N- wickets per innings at neutral,WI_W- wickets per innings in won matches,WI_L- wickets per innings in lost matches.

Earning point system [11].

We say how the team earn points in all three formats by the below tables from dream’11 concept.

1.1. Point system

Allocation of points for a player	T20	ODI	Test
If he is in the playing XI	2	2	2
If he scores one run	0.5	0.5	0.5
If he takes Wicket(including run-out)	10	12	8
If he takes a catch (per catch)	4	4	4
If he dismissed a player by Caught and bowled	14	16	12
If he does Stumping or Run out(Direct hit)	6	6	6
If he is helps to make Run out (involved)	4/2	4/2	4/2
If he Got out for zero run(batsmen, wicket-keeper &all-rounders)	-2	-3	-4

The captain of our fantasy cricket team is going to gain twice as much.

The Vice Captain will earn one and a half times more points for his performance.

Only strike rate scores below 70 runs per 100 balls are valid.

After the live match starts, our Fantasy team receives points based on the success of the players we pick. After the match is over, final scores, rankings and winners are declared.

1.2. Premium rating

Allocation of points for a player	T20	ODI	Test
If he scores boundary(per boundary)	0.5	0.5	0.5
If he hits six (per six)	1	1	1
If he hits half century	4	2	2
If he hits century	8	4	4
If he bowls a maiden over(run less over)	4	2	NA
If he takes 4 wickets in a match	4	2	2
If he takes 5 wickets in a match	4	2	2

1.3.Strike rate (Excluding Bowlers)

Allocation of points for a player	T20	ODI	Test
Applicable for players who bat minimum balls	10-balls	5-overs	NA
For every 100 balls if he scores around 60 to 70 runs	-1	-	NA
For every 100 balls if he scores around 50 to 59 runs per 100 balls	-2	-	NA
For every 100 balls if he scores around 40 to 49 runs per 100 balls	-3	-	NA
For every 100 balls if he scores around 30 to 39 runs per 100 balls	-	-1	NA
Less than 39 runs per 100 balls	-	-3	NA

1.4. Economy rate

Allocation of points	T20	ODI	Test
Applicable to players bowling minimum overs	2-overs	5-overs	NA
If a bowler gives around 6 to 5 runs per over	1	-	NA
If a bowler gives around 4.99 to 4 runs per over	2	-	NA
If a bowler gives Below 4 RPO	3	-	NA
If a bowler gives around 9 to 10 RPO	-1	-	NA
If a bowler gives around 10.1 to 11 RPO	-2	-	NA
If a bowler gives above 11 runs per over	-3	-	NA
If a bowler gives around 4.5 and 3.5 RPO	-	1	NA
If a bowler gives around 3.49 and 2.5 RPO	-	2	NA
If a bowler gives below 2.5 runs per over	-	3	NA
If a bowler gives around 7 and 8 RPO	-	-1	NA
If a bowler gives around 8.1 and 9 RPO	-	-2	NA
If a bowler gives above 9 runs per over	-	-3	NA

In this paper, we propose a new technique namely Best’11. A new methodology is provided to select a cricket team and Best’11 players from it. To justify this, we have considered cricinfo [3], dream ’11 [11] and howstat [4]. With this we have given an example of choosing 15 players for Indian team 2019 world cup. It is evident to note that our method gives 93 percent of accuracy in choosing teams & players and it is justified by means of an example.

This paper is arranged as follows. Section 2, gives the basic terminologies and historical background that are required in choosing Best’11 cricket players. Section 2 we introduced the new concept Best’11 assignment models from the inspiration of dream’11 [11]. Section 3 provides marked down algorithm for choosing Best’11 players using rough matrix theory derived from the information systems. Section 4 give MCDM approach using rough matrix theory in cricket. Section 5 dealt batting positions using assignment models for all three formats. Section 6 compares our approach with the existing approach [15]. The merits and demerits of our method is clearly exhibited in this section using evaluation index. A concluding remark is provided in Section 7.

2 Best’11-An overview

In this section, enlighted by the theory of Dream’11, rough sets and assignment models, we coined new concept of Best’11 game. Best’11 cricket is about using our experience and expertise in cricketing to build Best’11 squad. Our Best’11 squad receives points based on how our chosen cricketers are doing in matches in real life. It is a strategy game focused on choosing cricketers before starting a match. Fantasy cricket on Best’11 is a ability game so brush on your knowledge of cricket and you’ll have a much better chance of winning. Using our cricket thinking skills we can choose Best’11 and we are going to play tournaments with the positive intention of winning. If we win the match checks our teams score after the match.

To create a team before the match, we need to follow the following conditions,

i) Choose captain and vice-captain

ii) Pick the 11 cricketers from both the teams according to Definition 4.1.

iii) Add players to each category from the available selection.(i.e., you can add 5-batsman, 2-all-rounders, 1-wicketkeeper and 3-bowlers).

Step 1. Create your team

Use your skills to create your Best’11 cricket team with 1 billion in virtual money.

Step 2. Join a league

Enter a league and use the skills to beat other teams.

Step 3. Check your status

After the match, test our teams score to see if we’ve won the title.

Step 4. Creating our team

Choose a match first and set up a team by selecting 11 cricketers, including the captain and vice-captain.

Select 11 cricketers from the two squads.

Select players for our team on the basis of their ranking, i.e., ICC Ranking and Best’11Ranking.

Add players to each category from the available selection.

i.e., you can add 5-batsman, 2-all-rounders, 1-wicketkeeper and 3-bowlers.

Enable selection from one national team up to a maximum of 7 cricketers.

Select captain and vice-captain.

Governing the squad

Create limitless modifications to our team until the deadline.

Even we may change captain and vice captain.

Make sure you keep a eye on who’s playing and who isn’t, tracking your squad at all times and improve the chances of winning the cash prize.

3 Algorithm

Algorithm for Best’11 fantasy cricket game by using rough matrix theory is as follows.

Step 1.Choose the specific match that you want your Best’11 team to build for.

Step 2. From the list of probable squad of both the teams, we have to give Estimated Ranking Parameter for all the players, in conditional attributes.

Step 3.Find Estimated Ranking Parameter =

GLB{ICC Ranking, Best’11 Ranking}.

Step 4. Based on the conditional attributes, provide ‘1’ for selected players and ‘0’ for remaining players in the decision table.

Step 5. From the decision table, we have to form rough matrix.

Step 6. Based on the rough matrix, we have to select ‘11’ players from both the teams. The requirements given as 5 batsmen,3 bowlers,2 all rounders and 1 wicket keeper.

Step 7. Conclusion

3.1. Problem statement

Consider 32(14 batsmen, 10 bowlers, 4 wicket keepers and 4 all-rounders) players from 2 countries.The problem is to choose Best’11 players from both the team.

3.2. The data tables

We Compare the ICC ranking with our Best’11 ranking and provide in the form of 4 tables as below.

In following tables, T1BAT1 means Team 1 Batsmen number 1, similarly T2BAT8 means Team 2 Batsmen number 8.

3.2.1.Batsman information data table

Batsman	ICC Ranking			Best’11 Ranking
	Test	ODI	T20	Test	ODI	T20
T1BAT1	1	1	14	1	1	10
T1BAT2	43	9	11	50	5	9
T1BAT3	4	-	-	3	-	-
T1BAT4	15	43	-	18	50	-
T1BAT5	55	2	9	60	2	7
T1BAT6	34	-	6	40	-	5
T1BAT7	-	82	54	-	80	45
T2BAT8	72	18	3	60	15	3
T2BAT9	-	34	5-	-	28	6
T2BAT10	12	-	-	9	-	-
T2BAT11	39	62	-	35	54	-
T2BAT12	3	22	-	5	30	-
T2BAT13	6	6	52	8	9	56
T2BAT14	-	-	16	-	-	12

3.2.2.Wicket Keeper’s information data table

Wicket Keeper	ICC Ranking			Best’11 Ranking
	Test	ODI	T20	Test	ODI	T20
T1WK1	48	-	-	32	-	-
T1WK2	-	21	65	-	25	52
T2WK3	46	-	-	38	-	-
T2WK4	63	24	78	52	18	70

3.2.3. Bowler’s information data table

Bowlers	ICC Ranking			Best’11 Ranking
	Test	ODI	T20	Test	ODI	T20
T1BOWL1	28	1	21	15	1	16
T1BOWL2	31	24	20	35	16	18
T1BOWL3	21	64	0	21	50	-
T1BOWL4	6	-	-	5	-	-
T1BOWL5	52	3	3	45	3	2
T2BOWL6	15	17	70	12	15	50
T2BOWL7	8	19	-	9	16	-
T2BOWL8	7	-	-	6	-	-
T2BOWL9	-	73	5	-	68	4
T2BOWL10	9	10	-	8	8	-

3.2.4. All-Rounder information data table

All-Rounder	ICC Ranking			Best’11 Ranking
	Test	ODI	T20	Test	ODI	T20
T1AR1	28	18	32	25	15	28
T1AR2	2	16	-	2	14	-
T2AR3	31	12	98	28	10	60
T2AR4	-	17	59	-	15	52

4 MCDM approach using rough matrix theory in cricket

Definition 4.1. (Estimated Ranking Parameter)

We will define an Estimated Ranking Parameter(ERP). ERP =GLB{ICC Ranking, Best’11 Ranking}

4.1.2. Visualizing problems statement as information systems

Let R_S = {Universe(U), Attributes (A),

Conditional Attibutes(C), Decision(D)}, where U = {Bat{1,2, ...,14},Bowl{1,2, ...,10},

Wk{1,2,3,4},All-Rounder{1,2,3,4}}.

A = {Test, ODI, T20},

C = Estimated Ranking Parameter. (ERP)

D = {Yes, No}

4.2. Estimated Ranking Parameter and decision table for all rounders

All-Rounder	Estimated Ranking Parameter			Decision
	Test	ODI	T20	Test	ODI	T20
T1ALL1	25	15	28	Yes	No	Yes
T1ALL2	2	14	-	Yes	Yes	No
T2ALL3	28	10	60	No	Yes	No
T2ALL4	-	15	52	No	No	Yes

4.3. Estimated Ranking Parameter and decision table for wicket keepers

Wicket Keeper	Estimated Ranking Parameter			Decision
	Test	ODI	T20	Test	ODI	T20
T1WK1	32	-	-	Yes	No	No
T1WK2	-	21	52	No	No	Yes
T2WK3	38	-	-	No	No	No
T2WK4	52	18	70	No	Yes	No

4.4. Estimated Ranking Parameter and decision table for bowlers

Bowlers	Estimated Ranking Parameter			Decision
	Test	ODI	T20	Test	ODI	T20
T1BOWL1	15	1	16	No	Yes	Yes
T1BOWL2	31	16	18	No	No	No
T1BOWL3	21	50	-	No	No	No
T1BOWL4	5	-	-	Yes	No	No
T1BOWL5	45	3	2	No	Yes	Yes
T2BOWL6	12	15	50	No	No	No
T2BOWL7	8	16	-	Yes	No	No
T2BOWL8	6	-	-	Yes	No	No
T2BOWL9	-	68	4	No	No	Yes
T2BOWL10	8	8	-	No	Yes	No

4.5.Estimated Ranking Parameter and Decision table for batsmen

Batsman	Estimated Ranking Parameter			Decision
	Test	ODI	T20	Test	ODI	T20
T1BAT1	1	1	10	Yes	Yes	No
T1BAT2	43	5	9	No	Yes	Yes
T1BAT3	3	-	-	Yes	No	No
T1BAT4	15	43	-	No	No	No
T1BAT5	55	2	7	No	Yes	Yes
T1BAT6	34	-	5	No	No	Yes
T1BAT7	-	80	45	No	No	No
T2BAT8	60	15	3	No	Yes	Yes
T2BAT9	-	28	5	No	No	Yes
T2BAT10	9	-	-	Yes	No	No
T2BAT11	35	54	-	No	No	No
T2BAT12	3	22	-	Yes	No	No
T2BAT13	6	6	52	Yes	Yes	No
T2BAT14	-	-	16	No	No	No

4.6. Indiscernibility relation matrix from the above table

From the tables listed above, table columns are identified by player attributes and rows, while table entries are attribute values. Therefore each table row can be interpreted as details about a particular player. For example, in the above table T1BAT1 is defined by the following value collection of attributes (Test Ranking-1), (ODI Ranking-1), (T20 Ranking-1). So he was chosen in TEST-YES & ODI-YES, but T20-NO which forms the player knowledge. Likewise we can produce all the relationships of indiscernibility that shape the table above. We will construct Rough matrix from the table now. Matrix entries were represented as row wise in the following table, first 14 entries for batsmen, next 10 entries for bowlers, then 4 entries for wicket keepers and lastly 4 entries for all-rounders.

$R_{M}^{T} = \begin{matrix} Test \\ ODI \\ T 20 \end{matrix} [\begin{matrix} 1 \\ 1 \\ 0 \end{matrix} \begin{matrix} 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 1 & 1 & 0 & 0 \\ 1 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 1 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 1 & 0 & 1 & 1 & 0 \\ 1 & 0 & 0 & 1 & 1 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 1 \end{matrix}]$

T1BAT1	T1BAT1	T1BAT2
T1BAT3	T1BAT2	T1BAT5
T2BAT10	T1BAT5	T1BAT6
T2BAT12	T2BAT8	T2BAT8
T2BAT13	T2BAT13	T2BAT9
T1BOWL4	T1BOWL1	T1BOWL1
T2BOWL7	T1BOWL5	T1BOWL5
T2BOWL8	T2BOWL10	T2BOWL9
T1WK1	T2WK4	T1WK2
T1ALL1	T1ALL2	T1ALL1
T1ALL2	T2ALL3	T2ALL4

From the above matrix $R_{M}^{T}$ , we concluded the above 11 will be the Best’11 for three formats. In the above table first column players are selected for playing Best’11 game for test matches. Similarly, second column players selected for ODI’s and last Column players for T20’s.

5 Determining batting positions using assignment models

After selecting the Best 11 for all three formats, to increase the winning percentage, we want to choose one more criteria best position(batting order)for all the players based upon their averages in the respective formats using assignment models is discussed in this section. Here, we are going to assign a batting position for the players selected in Test format based on the batting average in different position by using assignment model problem. The following table entries are batting averages of players in different batting positions.

The following problem can be read as row wise for

(R1,T1BAT1),(R2,T1BAT3)(R3,T2BAT10),

(R4,T2BAT12),(R5,T2BAT13),(R6,T1BOWL4),

(R7,T2BOWL7),(R8,T2BOWL8),(R9,T1WK1),

(R10,T1ALL1),(R11,T1ALL2).

Solution

Step-I: Applying the row operation, i.e., subtracting the highest element of each row from every element of the respective row, we got the following reduced matrix

Step-II: Now applying the column operation, i.e., subtracting the smallest element of each column from every element of the respective column. We get the following reduced matrix.

Step-III:

Step-IV:

Step-V:

Step-VI:

Step-VII:

BATTING POSITION 1	T1BAT3	116
BATTING POSITION 2	T2BAT10	113
BATTING POSITION 3	T2BAT13	17
BATTING POSITION 4	T1BAT1	61
BATTING POSITION 5	T2BAT12	61
BATTING POSITION 6	T1WK1	46
BATTING POSITION 7	T1BOWL4	25
BATTING POSITION 8	T1ALL1	53
BATTING POSITION 9	T1ALL2	38
BATTING POSITION 10	T2BOWL7	26
BATTING POSITION 11	T2BOWL8	15
OPTIMAL SOLUTION	571

From step VII of the assignment problem, we can fix the batting position of the players in test format to win Best’11 game and also it provides us optimal runs obtained by this batting order is 571. we can implement the same idea to ODI and T20 to get opt batting position and optimal runs to their corresponding solutions.

6 Prediction of Indian world cup 2019 squad using our approach

In this section, a comparative analysis of our work with the existing literature taken from [15] is done. We introduce the idea of several indices to justify our work. So with respect to players data there are two main classifications 1) overall Performance, 2)performance during 2015-2019. Our goal is to pick 15 Indian squad members out of 24 for the 2019 World Cup.Our goal is to implement the new algorithm defined in Section 3 and methodology in Section 4 to select 15 member squad.

In Table 1, we can see batsmen data,which is predominantly dominant by 2015-19 performance and overall performance(starting from individual’s debut) has less dominant comparing to the prior. Our aim is to find new index from the data, which was dominated by their batting abilities. There are three indices that we are going to define using Table 1.

Table 1

Batsman’s information table (Overall & 2015-19)

Players	Span	Mat	Inns	NO	Runs	HS	Ave	BF	SR	100	50	IC _Bat	Bat I1	Rank
V Kohli	overal	248	239	39	11867	183	59.33	12726	93.25	43	58	66.11
	2015-2019	69	69	14	4306	160^*	78.29	4379	98.33	19	16	66.22	66.18	1
R Sharma	overall	224	217	32	9115	264	49.27	10250	88.92	29	43	56.48
	2015-2019	71	71	9	3790	208^*	61.12	3977	95.29	15	16	57.64	57.29	2
S Dhawan	overall	136	133	7	5688	143	45.14	6050	94.01	17	29	52.56
	2015-2019	67	67	4	2848	143	45.20	2921	97.50	8	15	50.43	51.07	3
A Rahane	overall	90	87	3	2962	111	35.26	3767	78.63	3	24	40.69
	2015-2019	36	34	1	1378	103	41.75	1718	80.20	1	15	42.46	41.93	7
A Rayudu	overall	55	50	14	1694	124^*	47.05	2143	79.04	3	10	44.13
	2015-2019	28	26	8	951	124^*	52.83	1159	82.05	2	5	46.65	45.89	5
M Pandey	overall	26	21	7	492	104^*	35.14	535	91.96	1	2	42.04
	2015-2019	23	18	6	440	104^*	36.66	479	91.85	1	2	42.62	42.45	6
S Iyer	overall	18	16	1	748	103	49.86	742	100.80	1	8	51.18
	2015-2019	6	5	0	210	88	42.00	218	96.33	0	2	45.90	47.48	4
MS Dhoni	overall	350	297	84	10773	183^*	50.57	12303	87.56	10	73	55.80
	2015-2019	79	61	16	2001	134	44.46	2461	81.30	1	13	43.67	47.31	1
D Karthik	overall	94	79	21	1752	79	30.20	2392	73.24	0	9	34.95
	2015-2019	20	17	8	425	64^*	47.22	565	75.22	0	2	41.65	39.64	4
KL Rahul	overall	32	31	5	1239	112	47.65	1423	87.06	4	7	46.68
	2015-2019	14	13	3	343	100^*	34.30	424	80.89	1	2	38.39	40.87	3
R Pant	overall	16	14	0	374	71	26.71	361	103.60	0	1	41.86
	2015-2019	5	4	0	93	36	23.25	71	130.98	0	0	48.59	46.58	2
V Shankar	overall	12	8	1	223	46	31.85	246	90.65	0	0	39.94
	2015-2019	9	5	0	165	46	33.00	171	96.49	0	0	42.15	41.48	3
H Pandya	overall	54	38	6	957	83	29.90	828	115.57	0	4	47.03
	2015-2019	54	38	6	957	83	29.90	828	115.57	0	4	47.03	47.03	2
K Jadhav	overall	73	52	19	1389	120	42.09	1367	101.60	2	6	48.32
	2015-2019	58	39	13	1154	120	44.38	1121	102.94	2	5	49.53	49.17	1
S Raina	overall	226	194	35	5615	116^*	35.31	6005	93.50	5	36	46.77
	2015-2019	11	10	0	227	53	22.70	258	87.98	0	1	35.57	38.93	4
R Jadeja	overall	165	110	38	2296	87	31.88	2671	85.96	0	12	39.74
	2015-2019	32	20	6	282	32	20.14	348	81.03	0	0	32.37	34.58	5
R Ashwin	overall	111	61	19	675	65	16.07	776	86.98	0	1	32.62
	2015-2019	15	6	1	22	15^*	4.40	29	75.86	0	0	24.52	26.95	7
A Patel	overall	38	20	6	181	38	12.92	190	95.26	0	0	33.75
	2015-2019	25	13	4	140	38	15.55	156	89.74	0	0	33.14	33.32	6
B Kumar	overall	114	52	15	526	53^*	14.21	703	74.82	0	1	28.23
	2015-2019	60	27	10	362	53^*	21.29	497	72.83	0	1	30.47	29.79	1
J Bumrah	overall	64	15	10	19	10^*	3.80	44	43.18	0	0	14.47
	2015-2019	49	10	5	18	10^*	3.60	42	42.85	0	0	14.30	14.35	6
Y Chahal	overall	52	9	3	49	18^*	8.16	84	58.33	0	0	20.76
	2015-2019	41	7	3	34	18^*	8.50	67	50.74	0	0	18.62	19.26	4
K Yadav	overall	60	21	12	118	19	13.11	188	62.76	0	0	24.07
	2015-2019	44	18	10	100	19	12.50	171	58.47	0	0	22.54	23.00	3
M Shami	overall	77	36	17	147	25	7.73	180	81.66	0	0	27.59
	2015-2019	16	8	3	22	8	4.40	27	81.48	0	0	26.20	26.62	2
U Yadav	overall	75	24	14	79	18^*	7.90	134	58.95	0	0	20.85
	2015-2019	27	9	3	35	18^*	5.83	65	53.84	0	0	18.48	19.19	5

Definition 6.1. We define batting index by cases is denoted as IC_Bat which is a combination of ICO_Bat and ICF_Bat. ICO_Bat is known as overall index by cases and ICF_Bat is known as Filtered (2015-19 performance) index by cases can be measured using formulae below

ICO_Bat =

(0.4) × O_Ave + (0.3)×O_SR + (0.2) × O₁₀₀ + (0.1) × O₅₀

ICF_Bat =

(0.4) × F_Ave + (0.3) × F_SR + (0.2) × F₁₀₀ + (0.1) × F₅₀

From the above two formulae we can find batting index 1(Bat I1) by the following formula

BatI1 = (0.3) × ICO_Bat + (0.7) × ICF_Bat.

Our assessment of a player’s recent performance has a strong impact on team success, so we gave 0.7 weight to recent performance (2015-2019) and 0.3 weight to overall results.

Table 2 reveals the Batsmen information table based on the venues and results. Using Table 2 our goal is to find Bat I2. In England, played the 2019 World Cup, which is a neutral venue for the Indian team. Therefore, instead of home venue, we need to give comparatively higher weights for away and neutral venue, and also give higher weights to matches won rather than lost.

Table 2

Batsmen Information table based on venues and results

Players	home	away	neutral	won	lost
Weights	0.15	0.25	0.3	0.2	0.1	Bat I2	Rank
V Kohli	73.85	79.00	129.00	97.16	48.08	93.77	1
R Sharma	59.14	56.03	88.71	70.91	48.41	68.51	2
S Dhawan	31.18	47.80	68.00	47.13	40.91	50.54	5
A Rahane	39.68	43.70	0.00	42.52	39.07	29.29	11
A Rayudu	41.66	65.75	43.75	72.45	4.80	50.78	4
M Pandey	20.12	90.33	8.00	60.00	14.40	41.44	8
S Iyer	54.00	24.00	0.00	61.00	13.50	27.65	12
MS Dhoni	39.33	60.86	24.00	62.55	31.00	43.92	7
D Karthik	65.50	37.00	48.66	154.50	12.00	65.77	3
KL Rahul	12.50	46.60	60.00	41.00	12.33	40.96	9
R Pant	23.25	0.00	0.00	0.00	25.33	6.02	20
V Shankar	30.00	45.00	0.00	45.50	24.66	27.32	13
H Pandya	37.83	18.08	35.75	31.76	27.80	30.05	10
K Jadhav	45.20	41.33	52.00	57.14	30.45	47.19	6
S Raina	13.60	31.80	0.00	30.33	19.42	18.00	17
R Jadeja	17.33	23.00	21.00	26.66	18.22	21.80	16
R Ashwin	16.00	1.66	1.00	0.00	1.40	3.26	22
A Patel	20.16	6.33	0.00	15.00	16.00	9.21	18
B Kumar	16.77	30.00	15.50	31.66	15.63	22.56	15
J Bumrah	4.25	0.00	1.00	10.00	2.00	3.14	23
Y Chahal	1.33	30.00	0.00	1.00	11.00	9.00	19
K Yadav	11.20	15.00	14.00	71.42	16.20	25.53	14
M Shami	9.50	1.00	0	3.00	4.75	2.75	24
U Yadav	15.50	1.00	0	4.00	6.20	4.00	21

Definition 6.2. Through the following definition, we define batting index 2 as

BatI2 = (0.15) × Ave_H + (0.25) × Ave_A + (0.3) × Ave_N

+ (0.2) × Ave_W + (0.1)×Ave_L.

In Table 3, we obtained Ranks based on OBatI.

Table 3

Decision Table based on OBatI

Player	Bat I1	Bat I2	OBatI	Rank
V Kohli	66.18	93.77	79.98	1
R Sharma	57.29	68.51	62.90	2
D Karthik	39.64	65.77	52.71	3
S Dhawan	51.07	50.54	50.81	4
A Rayudu	45.89	50.78	48.34	5
K Jadhav	49.17	47.19	48.18	6
MS Dhoni	47.31	43.92	45.62	7
M Pandey	42.45	41.44	41.94	8
KL Rahul	40.87	40.96	40.92	9
H Pandya	47.03	30.05	38.54	10
S Iyer	47.48	27.65	37.57	11
A Rahane	41.93	29.29	35.61	12
V Shankar	41.48	27.32	34.40	13
S Raina	38.93	18.00	28.47	14
R Jadeja	34.58	21.80	28.19	15
R Pant	46.58	6.02	26.30	16
B Kumar	29.79	22.56	26.18	17
K Yadav	23.00	25.53	24.27	18
A Patel	33.32	9.21	21.26	19
R Ashwin	26.95	3.26	15.10	20
M Shami	26.62	2.75	14.68	21
Y Chahal	19.26	9.00	14.13	22
U Yadav	19.19	4.00	11.59	23
J Bumrah	14.35	3.14	8.74	24

Definition 6.3. From the above two definitions, we can define overall batting index OBatI as $OBatI = \frac{(BatI 1 + BatI 2)}{2} .$

OBatI is shown in Table 3.

The higher you get OBatI, the better rated player that you are.

In Table 4, we can see Information table based on players bowling skills in overall matches and matches between 2015-2019. Our objective is to find DI_Bow, ICO_Bow, ICF_Bow and BowI from Table 4.

Table 4

Information table based upon bowling skills (Overall & 2015-19)

						Dismissal percent by batting order
Bowler	Span	Mat	Inns	Wkts	Ave	Econ	SR	W/I	Top	Middle	Tail	DI _Bow	ICO _Bow	BowI	Rank
H Pandya	Overall	54	53	54	40.64	5.56	43.8	1.02	37.09
	2015-2019	54	53	54	40.64	5.56	43.8	1.02	44.4	46.3	9.3	37.95	37.09	37.09	10
K Jadhav	Overall	73	42	27	37.77	5.15	43.9	0.64	36.11
	2015-2019	58	36	27	34.70	5.15	40.4	0.75	48.1	44.4	7.4	38.85	33.85	34.75	8
R Jadeja	Overall	165	161	187	36.57	4.89	44.7	1.16	34.84
	2015-2019	32	32	31	48.32	5.01	57.8	0.97	28.9	47.6	23.5	33.43	43.45	40.01	11
B Kumar	Overall	114	113	132	34.60	5.02	41.2	1.17	33.68
	2015-2019	60	59	72	34.97	5.31	39.4	1.22	47.7	25	27.3	36.81	33.29	33.45	6
M Shami	Overall	77	76	144	25.42	5.58	27.2	1.89	25.64
	2015-2019	16	16	26	30.19	5.31	34.0	1.63	39.6	37.5	22.9	35.63	29.56	28.00	4
J Bumrah	Overall	64	64	104	24.43	4.55	32.1	1.63	26.22
	2015-2019	49	49	85	22.15	4.51	29.4	1.73	33.7	31.7	34.6	33.28	24.51	25.19	2
K Yadav	Overall	60	58	104	26.16	5.11	30.6	1.79	26.71
	2015-2019	44	42	87	21.74	4.93	26.4	2.07	32.7	46.2	21.2	34.45	23.71	24.91	1
Y Chahal	Overall	52	51	91	25.83	5.07	30.5	1.78	26.36
	2015-2019	41	40	72	24.61	4.89	30.1	1.80	27.5	52.7	19.8	33.52	25.76	26.00	3
V Shankar	Overall	12	9	4	52.50	5.40	58.2	0.44	45.00
	2015-2019	9	8	2	94.00	5.61	100.5	0.25	25	50	25	32.50	74.2	62.57	13
S Raina	Overall	226	101	36	50.30	5.11	59.0	0.36	44.64
	2015-2019	11	10	4	57.00	5.56	61.5	0.40	28	55	17	33.90	48.07	46.70	12
R Ashwin	Overall	111	109	150	32.91	4.91	40.1	1.38	32.27
	2015-2019	15	14	17	40.58	5.36	45.4	1.21	33	48	19	34.70	36.91	35.06	9
A Patel	Overall	38	35	45	31.31	4.43	42.4	1.29	32.41
	2015-2019	25	25	29	34.79	4.43	47.0	1.16	35	47	18	35.20	35.17	34.07	7
U Yadav	Overall	75	73	106	33.63	6.01	33.5	1.45	30.91
	2015-2019	27	26	39	37.38	6.45	34.7	1.50	41	40	19	36.30	32.77	32.03	5

Definition 6.4. In this definition we are going to coin two more indices dominant by bowling skills $\frac{W}{I} = \frac{Wickets}{Innings}$ DI_Bow = (0.5) × (Top) + (0.3) × (Middle) + (0.2) × (Tail).

In the above calculation, we have given highest weight (0.5) to the bowler who dismissed opponents top order batsman(Batsman 1,2 and 3), higher weight (0.3) to the bowler who dismissed opponents middle order batsman (Batsman 4,5,6 and 7) and least weight (0.2) to the bowler who dismissed opponents tail enders(Batsman 8,9,10 and 11). If DI_Bow is high which means he dismissed more number of top order batsman on the other hand if some bowlers dismissal percent is quite similar in all three mode of dismissals which means the bowler is impact bowler and gives breakthrough whenever he ask to bowl by captain.

Players	Top	Middle	Tail	DI _Bow
J Bumrah	33.7	31.7	34.6	33.28
H Pandya	44.4	46.3	9.3	37.95

Definition 6.5. We define bowling index is as follows.

BowI = (0.4) × ICO_Bow + (0.6) × ICF_Bow

To find BowI, we need two values ICO_Bow and ICF_Bow which is defined by

ICO_Bow = (0.4) × O_Ave + (0.3) × O_SR + (0.2) × DI_Bow + (0.1) × O_W/I

ICF_Bow = (0.4) × F_Ave + (0.3) × F_SR + (0.2) × DI_Bow + (0.1) × F_W/I

In Table 5, we are given Bowlers Information table based on venues and results. Our objective is to find OBowI using this table.

Definition 6.6. Using Table 5, we define the following indices,

IVR_H = (0.4) × Ave_H + (0.4)×SR_H + (0.3)×WI_H

IVR_A = (0.4) × Ave_A + (0.4)×SR_A + (0.3)×WI_A

IVR_N = (0.4) × Ave_N + (0.4)×SR_N + (0.3)×WI_N

IVR_W = (0.4) × Ave_W + (0.4)× SR_W + (0.3)×WI_W

IVR_L = (0.4) × Ave_L + (0.4)×SR_L + (0.3)×WI_L

ICV_Bow =

(0.25) ×IVR_H + (0.35) ×IVR_A + (0.4) ×IVR_N

ICR_Bow = (0.6) ×IVR_W + (0.4)×IVR_L $ICF = \frac{({ICO}_{Bow} + {ICR}_{Bow})}{2}$ $OBowI = \frac{(ICF + {DI}_{Bow} + BowI)}{3}$ In Table 6, we rank the bowlers based on their OBowI obtained from Table 5 and we classify by their nature of bowling in Table 7. In Table 8, we assign rank for the batsmen based on OBatI and we classify by their nature in Table 9. Comparison and similarity percentage tables are given in Tables 10 & 11 respectively. We coined assignment problem in Table 12 and obtained solution in Table 13. In Table 14 we concluded our best batting position.

Table 5

OBowI index based on bowlers information table using venues and results in 2015–19

Players	Nature	Wt	Mat	Inns	Wkts	Ave	Econ	SR	W/I	IVR	ICF	15 to19	I by D	BowI	OBowI
H Pandya	home	0.25	18	18	21	34.61	5.76	36.0	1.17	28.48
	away	0.35	22	21	19	43.36	5.28	49.2	0.90	37.20
	neutral	0.4	14	14	14	46.00	5.70	48.3	1.00	37.92	35.31
	won match	0.6	38	38	41	37.39	5.47	41.0	1.08	31.57
	lost match	0.4	15	15	13	50.92	5.78	52.8	0.87	41.66	35.61	35.46	37.95	37.09	36.83
K Jadhav	home	0.25	23	15	10	46.00	5.54	49.8	0.67	38.45
	away	0.35	24	13	8	33.87	5.22	38.8	0.62	29.19
	neutral	0.4	11	8	9	22.88	4.38	31.3	1.13	21.90	28.59
	won match	0.6	44	27	23	30.65	4.90	37.5	0.85	27.43
	lost match	0.4	12	8	3	68.33	6.61	62.0	0.38	52.21	37.34	32.97	38.85	34.75	35.52
R Jadeja	home	0.25	11	11	14	38.21	4.87	47.0	1.27	34.34
	away	0.35	12	12	6	93.00	4.98	112.0	0.50	82.10
	neutral	0.4	9	9	11	36.81	5.25	42.0	1.22	31.77	50.03
	won match	0.6	17	17	19	38.42	4.51	51.1	1.12	36.03
	lost match	0.4	13	13	9	74.77	5.75	78.0	0.69	61.25	46.12	48.07	33.43	40.01	40.50
B Kumar	home	0.25	22	22	28	40.28	6.02	40.1	1.27	32.41
	away	0.35	28	27	31	32.80	5.03	39.0	1.15	28.95
	neutral	0.4	10	10	13	28.69	4.41	39.0	1.30	27.34	29.17
	won match	0.6	40	40	56	26.71	4.77	33.5	1.40	24.36
	lost match	0.4	19	19	16	63.87	6.36	60.2	0.84	49.80	34.54	31.85	36.81	33.45	34.04
M Shami	home	0.25	7	7	8	51.87	6.01	51.7	1.14	41.66
	away	0.35	9	9	18	20.55	4.70	26.2	2.00	19.10
	neutral	0.4	0	0	0	0.00	0.00	0.0	0.00	0.00	17.10
	won match	0.6	10	10	22	21.63	5.36	24.1	2.20	18.73
	lost match	0.4	5	5	3	83.33	5.10	98.0	0.60	72.65	40.30	28.70	35.63	28.00	30.78
J Bumrah	home	0.25	25	25	39	28.00	4.97	33.7	1.56	24.99
	away	0.35	15	15	34	13.32	3.73	21.3	2.27	14.30
	neutral	0.4	9	9	12	28.16	4.39	38.4	1.33	26.89	22.01
	won match	0.6	38	38	71	19.16	4.27	26.9	1.87	18.80
	lost match	0.4	11	11	14	37.28	5.28	42.3	1.27	32.09	24.11	23.06	33.28	25.19	27.18
K Yadav	home	0.25	16	15	30	26.56	5.80	27.4	2.00	21.98
	away	0.35	22	21	47	18.25	4.57	23.9	2.24	17.31
	neutral	0.4	6	6	10	23.70	4.08	34.8	1.67	23.73	21.05
	won match	0.6	29	29	65	18.58	4.54	24.5	2.24	17.68
	lost match	0.4	12	11	17	34.05	5.96	34.2	1.55	27.61	21.65	21.35	34.45	24.91	26.90
Y Chahal	home	0.25	14	13	22	29.63	5.38	33.0	1.69	25.39
	away	0.35	22	22	44	20.97	4.68	26.8	2.00	19.51
	neutral	0.4	5	5	6	32.83	4.47	44.0	1.20	30.97	25.56
	won match	0.6	31	31	67	19.16	4.48	25.6	2.16	18.34
	lost match	0.4	9	8	4	106.25	6.42	99.2	0.50	82.28	43.91	34.74	33.52	26.00	31.42
V Shankar	home	0.25	5	4	2	55.00	6.28	52.5	0.50	43.10
	away	0.35	4	4	0	0	4.87	0	0.00
	neutral	0.4	0	0	0	0	0.00	0	0.00	0.00	10.78
	won match	0.6	6	6	2	57.50	5.60	61.5	0.33	47.67
	lost match	0.4	3	2	0	0	5.61	0	0.00	0.00	28.60	19.69	32.5	62.57	38.25
S Raina	home	0.25	5	4	1	87.00	6.21	84.0	0.25	68.45
	away	0.35	6	6	3	47.00	5.22	54.0	0.50	40.50
	neutral	0.4	0	0	0	0	0.00	0	0.00	0.00	31.29
	won match	0.6	4	3	3	23.66	5.46	26.0	1.00	20.06
	lost match	0.4	7	7	1	157.00	5.60	168.0	0.14	130.03	64.05	47.67	33.9	46.70	42.75
R Ashwin	home	0.25	4	4	4	50.50	6.37	47.5	1.00	39.40
	away	0.35	8	7	12	26.75	4.72	34.0	1.71	24.64
	neutral	0.4	3	3	1	167.00	5.75	174.0	0.33	136.47	73.06
	won match	0.6	7	7	10	33.50	5.07	39.6	1.43	29.53
	lost match	0.4	7	7	7	50.71	5.66	53.7	1.00	41.96	34.50	53.78	34.7	35.06	41.18
A Patel	home	0.25	13	13	13	44.53	4.94	54.0	1.00	39.61
	away	0.35	12	12	16	26.87	3.90	41.3	1.33	27.53
	neutral	0.4	0	0	0	0	0.00	0	0.00	0.00	19.54
	won match	0.6	19	19	25	27.68	3.99	41.6	1.32	27.98
	lost match	0.4	6	6	4	79.25	5.87	81.0	0.67	64.23	42.48	31.01	35.2	34.07	33.43
U Yadav	home	0.25	11	11	17	37.35	6.61	33.8	1.55	28.77
	away	0.35	14	13	19	38.21	6.44	35.5	1.46	29.78
	neutral	0.4	2	2	3	32.33	5.59	34.6	1.50	27.07	28.44
	won match	0.6	13	13	18	36.05	6.52	33.1	1.38	27.94
	lost match	0.4	12	12	20	36.55	6.27	34.9	1.67	28.91	28.33	28.39	36.3	32.03	32.24

Table 6

Players rank by bowling skills

Players	Index	Nature	Rank
K Yadav	26.90	Spin	1
J Bumrah	27.18	Pace	1
M Shami	30.78	Pace	2
Y Chahal	31.42	Spin	2
U Yadav	32.24	Pace	3
A Patel	33.43	Spin Bat	1
B Kumar	34.04	Pace Bat	1
K Jadhav	35.52	Bat Spin	1
H Pandya	36.83	Bat Pace	1
V Shankar	38.25	Bat Pace	2
R Jadeja	40.50	Spin Bat	2
R Ashwin	41.18	Spin	3
S Raina	42.75	Bat Spin	3

Table 7

Bowling styles & ranking of players dominance by bowling

Pace	Spin	Pace Bat	Bat Pace	Spin Bat	Bat Spin
J Bumrah	K Yadav	B Kumar	H Pandya	R Jadeja	K Jadhav
M Shami	Y Chahal		V Shankar	A Patel	S Raina
U Yadav	R Ashwin

Table 8

Players rank & OBatI dominance by batting skills

Player	Bat I1	Bat I2	OBatI	Style &Rank
V Kohli	66.18	93.77	79.98	Bat	1
R Sharma	57.29	68.51	62.90	Bat	2
S Dhawan	51.07	50.54	50.81	Bat	4
A Rahane	41.93	29.29	35.61	Bat	12
A Rayudu	45.89	50.78	48.34	Bat	5
M Pandey	42.45	41.44	41.94	Bat	8
S Iyer	47.48	27.65	37.57	Bat	11
MS Dhoni	47.31	43.92	45.62	Bat WK	7
D Karthik	39.64	65.77	52.71	Bat WK	3
KL Rahul	40.87	40.96	40.92	Bat WK	9
R Pant	46.58	6.02	26.30	Bat WK	16
V Shankar	41.48	27.32	34.40	Bat Pace	13
H Pandya	47.03	30.05	38.54	Bat Pace	10
K Jadhav	49.17	47.19	48.18	Bat Spin	6
S S Raina	38.93	18.00	28.47	Bat Spin	14
R Jadeja	34.58	21.80	28.19	Spin Bat	15
R Ashwin	26.95	3.26	15.10	Spin	20
A Patel	33.32	9.21	21.26	Spin Bat	19
B Kumar	29.79	22.56	26.18	Pace Bat	17
J J Bumrah	14.35	3.14	8.74	Pace	24
Y Y Chahal	19.26	9.00	14.13	Spin	22
K Yadav	23.00	25.53	24.27	Spin	18
M Shami	26.62	2.75	14.68	Pace	21
U Yadav	19.19	4.00	11.59	Pace	23

Table 9

Batting style & ranking dominance by Batting

Pace	Spin	Pace Bat	Bat Pace	Spin Bat	Bat Spin	Bat	Bat WK
M Shami	K Yadav	B Kumar	H Pandya	R Jadeja	K Jadhav	V Kohli	MS Dhoni
U Yadav	R Ashwin		V Shankar	A Patel	S Raina	R Sharma	KL Rahu
J J Bumrah	Y Y Chahal					S Dhawan	R Pant
						D Karthik
						A Rayudu
						M Pandey
						S Iyer
						A Rahane
3	3	1	2	2	2	8	3

Table 10

Comparison table for our approach against BCCI list

	Pace	Spin	Pace Bat	Bat Pace	Spin Bat	Bat Spin	Bat	Bat WK
All Players	3	3	1	2	2	2	8	3
Possible picks	Pace	Spin	Pace Bat	Bat Pace	Spin Bat	Bat Spin	Bat	Bat WK
Set 1	2	2	1	2	1	1	5	1
Selected	J Bumrah	K Yadav	B Kumar	H Pandya	R Jadeja	K Jadhav	V Kohli	MS Dhoni
list	M Shami	Y Chahal		V Shankar			R Sharma
for set 1							S Dhawan
from our							D Karthik
approach							A Rayudu
Possible picks	Pace	Spin	Pace Bat	Bat Pace	Spin Bat	Bat Spin	Bat	Bat WK
Set 2	2	2	1	1	2	2	4	1
Selected	J Bumrah	K Yadav	B Kumar	H Pandya	R Jadeja	K Jadhav	V Kohli	MS Dhoni
list for set 2	M Shami	Y Chaha				A Patel	S Raina	R Sharma
from our							S Dhawan
approach							D Karthik
Possible picks	Pace	Spin	Pace Bat	Bat Pace	Spin Bat	Bat Spin	Bat	Bat WK
Set 3	3	3	1	1	1	2	3	1
Selected	M Shami	K Yadav	B Kumar	H Pandya	R Jadeja	K Jadhav	V Kohli	MS Dhoni
list for set 3	U Yadav	R Ashwin				S Raina	R Sharma
from our	J J Bumrah	Y Y Chahal					S Dhawan
approach
Selected	J Bumrah	K Yadav	B Kumar	H Pandya	R Jadeja	K Jadhav	V Kohli	MS Dhoni
list from	M Shami	Y Chahal		V Shankar			R Sharma	KL Rahul
BCCI							S Dhawan
							D Karthik

Table 11

Similarity percentage for our approach with BCCI selected list

Set 1	93.33
Set 2	86.67
Set 3	80

Table 12

Assignment table to decide batting order

Position	1	2	3	4	5	6	7	8	9	10	11
V Kohli	37	17	63	55	32	23	13	M	M	M	M
R Sharma	56	65	15	31	45	29	14	M	M	M	M
MS Dhoni	M	49	83	57	50	47	45	17	M	M	M
S Dhawan	10	48	M	M	M	M	M	M	M	M	M
KL Rahul	54	43	26	17	73	M	M	M	M	M	M
K Jadhav	M	M	M	9	38	50	28	M	M	M	M
H Pandya	M	M	M	32	25	19	33	30	M	M	M
B Kumar	M	M	M	M	M	M	10	20	12	2	M
K Yadav	M	M	M	M	M	M	M	M	11	13	M
Y Chahal	M	M	M	M	M	M	M	M	3	11	7
J Bumrah	M	M	M	M	M	M	M	M	M	11	2

Table 13

Final solution table

Players	1	2	3	4	5	6	7	8	9	10	11
V Kohli	10	39	0	[0]	24	33	43	M	M	M	M
R Sharma	[0]	0	57	33	20	36	51	M	M	M	M
MS Dhoni	M	27	[0]	18	26	26	29	28	66	M	M
S Dhawan	29	[0]	M	M	M	M	M	M	M	M	M
KL Rahul	10	30	54	55	[0]	M	M	M	M	M	M
K Jadhav	M	M	M	40	12	[0]	22	M	M	M	M
H Pandya	M	M	M	0	8	14	[0]	0	M	M	M
B Kumar	M	M	M	M	M	M	M	[0]	0	10	11
K Yadav	M	M	M	M	M	M	M	M	[0]	0	M
Y Chahal	M	M	M	M	M	M	M	M	11	5	[0]
J Bumrah	M	M	M	M	M	M	M	M	M	[0]	0

Table 14

Best’11 batting order

Players	Batting Order
R Sharma	1
S Dhawan	2
MS Dhoni	3
V Kohli	4
KL Rahul	5
K Jadhav	6
H Pandya	7
B Kumar	8
K Yadav	9
Y Chahal	10
J Bumrah	11

Conclusion

Thus we have provided a Best’11 algorithm for choosing best 15 squad members. To justify our algorithm we have to compare our work with the existing literature namely [15]. We have concluded that our idea of choosing 15 squad members almost coincides (93 percent) with BCCI’s list. We have used assignment model and rough matrix theory to enhance our new algorithm.

As a further work, we plan to extend this idea for Test and T20I’s.

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