[math]u[/math]

[math]Z(u) = Z_{0}[1 - exp(-bu)][/math]

[math]Z_{0}[/math]

[math]b[/math]

[math]Z_{0}[/math]

[math]Z_{0}exp(-bu)[/math]

[math]u[/math]

[math]Z_{0}[/math]

[math]b[/math]

[math]Z_{0}(w)[/math]

[math]w[/math]

[math]b(w)[/math]

[math]10-w[/math]

[math]Z_{0}(w)[/math]

[math]b(w)[/math]

[math]w[/math]

[math]Z(N,0)[/math]

[math]u[/math]

[math]w[/math]

[math]Z(u,w)[/math]

[math]P(u,w) = \frac{Z(u,w)}{Z(N,0)}[/math]

[math]u_{1}[/math]

[math]w[/math]

[math]u_{2}[/math]

[math]u_{1} - u_{2}[/math]

[math]P(u_{1},w) - P(u_{2},w)[/math]

[math]R = 1 - (P(u_{1},w) - P(u_{2},w)) [/math]

[math]T[/math]

[math]RT[/math]

[math]n[/math]

[math]w[/math]

[math]R_{u} = 1 - P(N-x,w)[/math]

[math]R_{u}T[/math]

[math]u_{1}[/math]

[math]u_{2}[/math]

[math]R_{1} = 1 - (P(u_{1},w) - P(u_{2},w))[/math]

[math]R_{2}[/math]

[math]R_{1}[/math]

[math]R_{2} - R_{1}[/math]

[math]N[/math]

[math]N[/math]

[math] T_{Team2} = S_{Team1} + Avg(N)[R_{2} - R_{1}][/math]

The procedure for setting a revised target, which is the same for any number of stoppages at any stage of the match, is as follows.

For each team's innings

(a) from the table note the resource percentage the team had available at the start of their innings;

(b) using the table, calculate the resource percentage lost by each interruption;

(c) hence calculate the resource percentage available. If Team 2 have less resources available than Team 1, then calculate the ratio of the resources available to the two teams. Team 2's revised target is obtained by scaling down Team 1's score by this ratio. If Team 2 have more resources available than Team 1, then calculate the amount by which Team 2's resource percentage exceeds Team 1's. Work out this excess as a percentage of the average 50 over score and this gives the extra runs to add on to Team 1's score to give Team 2's target.

The Duckworth-Lewis method is based on the concept of batting resources, and calculates targets by taking into account the remaining resources after an interruption.The method was first proposed in the paper' by FC Duckworth and AJ Lewis, in thein 1998.While formulating the method, Duckworth and Lewis (hereafter DL), took into account certain stipulations that their new method must adhere to. It must be almost equally fair to both sides, furnish realistic targets that are independent of the first team's scoring pattern (as it is in normal games), and it should be easy to apply and comprehend.The method begins by recognizing that the side batting has with it two resources that can be quantified at any point in the innings: the overs remaining, and the wickets in hand. The ability of team to score is directly dependent on these two resources.At the point of interruption, the aim of the method is to basically reset the target score based on the change in these resources for the chasing team.For this, DL quantify the relationship between the runs that can be scored with the set of available resources.They begin with the expression for the runs that can be made inovers:Here,are the hypothetical average runs a team can score given infinite overs. This factor is calculated using average ODI scoring rates. The factoris the exponential decay factor that decays the runs scored, scaling it down based on the number of overs available. So, you can see, in infinite overs a team would scoreruns, but this is reduced by a difference ofin the case ofovers remaining.This expression is now to be modified to include the effect of having lost some number of wickets. For this, DL simply modify the factorsandto include the effect of wickets.We now have, the runs scored in infinite overs if you havewickets down. Similarly, we have, the decay factor in the case of havingwickets left. This makes sense: the number of runs you can score get affected by the wickets left. The logical assumption here is that more the number of wickets you have left, the more resources you can make, and thus, greater the runs you can make from that point on.Bear in mind, that the two essential functions here,and, are empirically calculated after computing scores at different points of hundreds of ODI matches. They are then fit into analytical forms that give sensible values and smooth derivatives for all values of. The full form of both these functions is not made public, and they keep getting updated.Here is a graph of the runs scored as a function of the two parameters from the DL paper:The current tables and graphs might be different, as this is from a 1998 paper. You can clearly see the graphs mimicking expected behaviour. The less the wickets lost, the more the runs you are expected to score for the same number of overs remaining.With this done, we can calculate theto be scored when a certain number of overs remain and a certain number of wickets are down.The expected score at the start of an innings is (N overs and 0 wickets down):After facingovers and withwickets down, this comes down to:The proportion of runs to be scored is:This is the central number in setting revised scores, as we shall see.Now, let us begin with the case where there is an interruption in the innings of the chasing side.They haveovers remaining and arewickets down when play is stopped. When they return, they now only haveovers left. As a result of this, they have lostovers.The run-scoring resources they have lost as a result are:. So now, the resources they have left are:If they were chasingruns, theiris simply:runs. This makes sense, because we multiply the target by the proportion of resources lost due to the interruption.Similarly, we can also compute theat every point in the innings. In case the innings stops at that point, the par score monitors the target of the chasing team.Ifovers have been bowled, and the team chasing iswickets down, they have used a proportionof their resources.Since they have used this much, the scoreby this point is simply their target scaled by the resources used:In a different set of cases, sometimes overs are lost in the innings of the team batting first, and then the chasing team is given the same number of overs as the setting team.In this instance, the DL method recognizes that overs lost at different points in the innings have different impacts on the scoring ability of Team 1. They factor this in, and provide a method to set revised scores for Team 2.The method recognizes that Team 1 loses coring opportunities through unplanned interruptions, and thus it is only fair to scale the target for Team 2 to take this into account.Suppose Team 1 stops withovers left and comes back to bat withovers left. They now have:as the proportion of resources remaining.At the beginning of their innings, with some reduced number of overs, let us say Team 2 hasresources remaining.The revised target is now set by comparing the two teams' resources available.If the two are equal, then the target for Team 2 is simply the same as Team 1's final score. If the resources available to Team 2 are less than, then a simple scaling is done by multiplying Team 1's final score with the ratio of the resources.DL find in their paper that if the resources for Team 2 are greater than those of Team 1, the scaling method often leads to unrealistic targets. However, they agree that the target should be greater than Team 1's final score.To resolve this with an easy-to-use method, DL multiply the difference of resourcesby the average score of all innings ofovers, whereis the complete number of overs of the uninterrupted match. They then add this value to the final score of Team 1.Phew!The following is an excerpt from a document by the Northwich Cricket Club, as a summarized guide for applying the DL method:They also have worked examples for multiple cases.For applying this method, you can find multiple online calculators, and even an Android App.