Analytical Evaluation of Performance of Cricket Squad by ANP-DEA ^†

Abstract

Different sports have different fan bases. In addition to this, there is a lot of craze, enthusiasm, and zeal among people, mainly youngsters. Cricket is one among them that has tremendous popularity not only among youngsters, but among all age groups. This creates a kind of pressure among team members to perform well in every game, as well as on the selectors to select the best players for the opening, middle orders, wicketkeeping, and bowling from the pool of players. A game cannot be won by a single player or by openers, among others; rather, it is a collective effort of all the members of the team. As such, it is necessary and one of the most important tasks to choose the players wisely so that they play well in their respective position. In this study, we try to formulate a model using MCDM (multicriteria decision-making techniques), which evaluates not only the performance of the players, but also the performance of different sets (i.e., openers, middle orders, wicket-keepers, and bowlers). For this, we propose a novel ANP-DEA (analytic network process–data envelopment analysis) technique and evaluate the best and worst performing set and their performance evaluation. A case study is conducted to properly visualize the proposed model.

1. Introduction

Cricket is one of the most popular sports in the world. It originated in England in the 16th century and become its national game. This game is not as popular in Europe as it is in other countries, such as India, Sri Lanka, the West Indies, and Australia. This game does not just comprise of a single individual; instead, it is basically a game of many people, including players, authorities, and selection committees [1,2]. Choosing a team is one of the challenging tasks for higher authorities because there should be a proper balance and trade-off between the players, whether they are batsmen, bowlers, or wicket-keepers. To earn a place in the playing eleven, each player should do their best. The authorities can only choose the 11 players for the team, so another 3–4 members are chosen as the supporting members so that if any player gets injured, they have a chance to play and provide support to the team in the absence of players [3]. Choosing a team is a cumbersome task that depends on many factors that are not only related to the physical health of the players, but also their mental health. Selection boards such as the ICC (International Cricket Conference) implemented the new ways and programs, which help in the development of teams and make them suitable for playing national and international games [4,5]. There is a good combination of different types of players coming from different states with different abilities, which contributes to the performance of the team [6]. While playing the game of cricket, there are basically different sets of players that are playing, which come under openers, middle orders, wicket-keepers, and bowlers [7,8], and proper trade-off between these players is necessary for team management [9,10]. The selection of players within these sets is a cumbersome task that basically involves a lot of different criteria, such as the number of matches played, runs scored, wickets taken, batting and bowling strike rate, and other criteria [11]. As such, in view of catering to all the criteria and for proper analysis of the problem, we use multicriteria decision-making techniques here. In this paper, we use the hybrid model ANP-DEA. The ANP (analytical network process) is basically an extension of the AHP (analytical hierarchy process), which was developed by Saaty in 1980 [12]. In the ANP, we use a network in place of an element hierarchy. As the interdependency of criteria is not considered in the AHP [13,14] or many other MCDM techniques, the main reason for using ANP in real-life scenarios is because all the issues are somehow interlinked [15,16]. To rank the DMUs (Decision Making Units), here we can say players, we are using DEA [17,18], the non-parametric technique for performance evaluation based on multiple inputs and outputs [19,20]. There are different models of DEA based on the need of the users, i.e., input-oriented or output-oriented. Moreover, through DEA, we not only recognize efficient DMUs [21,22], but we are also told how to make inefficient DMUs close to efficient DMUs [23,24] and make recommendations based on this. In this paper, we dealing with how to choose the best set of players for a team by taking into account the different criteria. As such, here we first analyze each set, i.e., the openers, middle orders, wicket-keepers, and bowlers, using the ANP based on the different criteria and the importance of criteria according to experts of cricket. Additionally, based on the weights that we obtain from the ANP, we analyze the importance of each set and then finally find out the optimal players that may be selected for playing in the team by using the DEA ranking scheme.

2. Methodology

First, we need to find the sets that contribute more in terms of other sets that we described above using the ANP. Here, we take five criteria, i.e., number of matches, batting innings, bowling innings, catches taken, and stumps made, and four sub-criteria that belong to the batting innings criterion and four sub-criteria that belong to the bowling innings criterion. The batting innings sub-criteria include not out, runs scored, batting average, and batting strike rate. The bowling innings sub-criteria include maiden bowled, wickets taken, bowling economy rate, and bowling strike rate. The prototype of the ANP diagram is shown in Figure 1. To carry out the pair-wise comparison, we designed a questionnaire that was filled by experts who conducting doing research in this area. Different scores were assigned by the researchers based on the Saaty scale as shown in

Table 1. Description of Saaty Scale.

Intensity	Description	Scale Value
Equal	A is equally important as B.	1–2
Moderate	A is a little more important than B.	3–4
Strong	A is more important than B.	5–6
Very Strong	A is very much more important than B.	7–8
Extreme	A is extremely important than B.	9

.

The algorithm for finding the best set of squads using the ANP is as follows:

• The weights of the criteria are determined without considering the interdependency between them, and the weight of the criterion matrix is represented by J₁.
• The weights of the sub-criteria are determined without considering the interdependency between them, and the weight of the sub-criterion matrix is represented by J₂.
• Now, interdependency among criteria is introduced. The weight of the criteria relative to each criterion is represented by J₃, and the final priority of the criteria is given by J₅ = J₁ * J₃.
• Similarly, interdependency among sub-criteria is introduced. The weight of the sub-criteria relative to each sub-criterion is represented by J₄, and the final priority of the criteria is given by J₆ = J₂ * J₄.
• The final priorities of the criteria are given by the multiplication of matrix J₅ * J₆.
• The weights of the alternatives are given by the sum of the final priorities of each alternative that we obtain in Step 6.

The results that we obtain from the selection of the best set based on the criteria and sub-criteria discussed above are as follows:

Table 2. Local and global weights of the criteria.

Criteria	Sub-Criteria	Local Weights	Global Weights
Matches		0.053	0.053
Batting innings	Not out	0.06	0.01728
	Runs scored	0.146	0.042048
	Batting avg.	0.45	0.1296
	Batting strike rate	0.342	0.098496
Bowling innings	Maiden bowled	0.056	0.006608
	Wickets taken	0.169	0.019942
	Bowling economy rate	0.429	0.050622
	Bowling strike rate	0.344	0.040592
Catches taken		0.201	0.201
Stumps made		0.338	0.338

shows the local and global weights of the criteria and sub-criteria;

Table 3. Interdependency matrix.

	Matches	Not out	Runs Scored	Batting Avg.	Batting Strike Rate	Maiden Bowled	Wickets Taken	BCR	BSR	Catches Taken	Stumps Made
Middle orders	0.086	0.506	0.542	0.3	0.246	0.158	0.145	0.161	0.144	0.375	0.186
Wicket-keepers	0.172	0.233	0.121	0.139	0.492	0.128	0.098	0.12	0.116	0.315	0.08
Openers	0.434	0.213	0.28	0.484	0.215	0.086	0.067	0.067	0.056	0.207	0.624
Bowlers	0.307	0.046	0.055	0.075	0.046	0.626	0.688	0.65	0.682	0.1	0.107

shows the interdependency matrix; and

Table 4. Final priority matrix.

	Matches	Not out	Runs Scored	Batting Avg.	Batting Strike Rate	Maiden Bowled	Wickets Taken	BCR	BSR	Catches Taken	Stumps Made
Middle orders	0.005	0.009	0.023	0.039	0.024	0.001	0.003	0.008	0.006	0.075	0.063
Wicket-keepers	0.009	0.004	0.005	0.018	0.048	0.001	0.002	0.006	0.005	0.063	0.027
Openers	0.023	0.004	0.012	0.063	0.021	0.001	0.001	0.003	0.002	0.042	0.211
Bowlers	0.016	0.001	0.002	0.010	0.005	0.004	0.014	0.033	0.028	0.020	0.036

and

Table 5. Weights of Alternatives.

	Weights
Middle orders	0.255376
Wicket-keepers	0.188643
Openers	0.382447
Bowlers	0.16834

show the final priority matrix, as well as the weights of the alternative.

From Table 5, we analyze that the set of openers is the best set. Now, to find the efficiency of the openers, we take a dataset from IPL 2019 of Chennai Super king teams and evaluate the 17 players. Using DEA, we try to find out the players that are a good fit for the team as openers. To do so, we are using the CCR (Charnes-Cooper-Rhodes) input-oriented model, which is given by the following equations:

$$\mathrm{Max} {\mathrm{E}}_{\mathrm{i}}={\sum }_{m=1}^n{\theta }_m{O}_{md}$$

(1)

$${{\displaystyle \sum }}_{i=1}^s{\lambda }_{i }{I}_{id}=1, i=1, 2\dots s$$

(2)

$${{\displaystyle \sum }}_{m=1}^n{\theta }_m{O}_{mu} <={{\displaystyle \sum }}_{i=1}^s{\lambda }_{i }{I}_{iu}, u=1, 2\dots k$$

(3)

$${\theta }_m \mathrm{and} {\lambda }_{i } >=0$$

(4)

Here, ${\theta }_m$ and ${\lambda }_{i }$ are the weights associated with the outputs and inputs, respectively; m represents the no. of outputs (m = 1, 2…n); i represents the number of inputs (i = 1, 2…s); d represents the DMU taken for evaluation; and u represents the number of DMUs (u = 1, 2…k).

In this study, the number of DMUs are 17. Here, the number of inputs taken are four, namely matches played by the players, batting innings of the players, how many times they returned to the pavilion not out, and balls faced by them; meanwhile, the number of outputs are also four, namely runs scores by the players, batting average, batting strike rate, and catches taken by the players. After evaluation based on the model discussed above, we obtain the results as shown in Figure 2.

From Figure 2, we can conclude that DMUs with an efficiency score equal to 1 are regarded as efficient openers and DMUs with an efficiency score of less than 1 are efficient openers. Peers are considered the benchmark for insufficient openers, e.g., for KM Jadhav, the peers are SR Watson, MS Dhoni, and SN Thakur. As such, we can select from efficient openers.

3. Conclusions

Team selection is one of the important tasks but, besides this, selection of players among a particular set is also important. The winning of the match also depends on teamwork and not only on a single player. As such, each player should contribute as much as possible and should also work on themselves by training properly so that they have a chance to play for the state, country, or any franchise. The mathematical formulation of the real-life problem is important as it is clear and crisp. Here, we explored different sets of players and then chose the best set. Among the best, we chose the best openers. Here, using the ANP by discarding the disadvantages of the AHP, we compared the different sets and found out that the set containing openers contribute more to team wins according to the data received by the experts; moreover, when we look deeper into the openers set, we find the efficient and inefficient openers by using DEA and the CCR input-oriented model. For future work, we can consider other inputs too, which are not considered in this research due to the limitation of data, and we can also explore different models of DEA.