Since the 1970s that the idea of a general algorithmic framework, which can be applied with relatively few modifications to different optimization problems, emerged. Metaheuristics: methods that combine rules and randomness while imitating natural phenomena. These methods are from now on regularly employed in all the sectors of business, industry, engineering. besides all of the interest necessary to application of metaheuristics, occasionally a new metaheuristic algorithm is introduced that uses a novel metaphor as guide for solving optimization problems. 2 League Championship Algorithm: A new algorithm for numerical function optimization By: Dr. A. H. Kashan

Someexamples Some examples particle swarm optimization algorithm (PSO): models the flocking behavior of birds; harmony search (HS): models the musical process of searching for a perfect state of harmony; bacterial foraging optimization algorithm (BFOA): models foraging as an optimization process where an animal seeks to maximize energy per unit time spent for foraging; artificial bee colony (ABC): models the intelligent behavior of honey bee swarms; central force optimization (CFO): models the motion of masses moving under the influence of gravity; imperialist competitive algorithm (ICA): models the imperialistic competition between countries; fire fly algorithm (FA): performs based on the idealization of the flashing characteristics of fireflies. 3 League Championship Algorithm: A new algorithm for numerical function optimization By: Dr. A. H. Kashan

4 Metaheuristics Evolutionary algorithms Trajectory methods Social, political, music, sport, etc Are inspired by nature’s capability to evolve living beings well adapted to their environment Evolution strategies Genetic programming Genetic algorithm Swarm intelligence Tabu search Variable neighborhood search Ant colony optimization Particle swarm optimization Artificial bee colony Bacterial foraging optimization Group search optimizer Harmony search Society and civilization Imperialist competitive algorithm League championship algorithm work on one or several neighborhood structure(s) imposed on the members of the search space. work on one or several neighborhood structure(s) imposed on the members of the search space. Any attempt to design algorithms or distributed problem-solving devices inspired by the collective behavior of social insect colonies and other animal societies League Championship Algorithm: A new algorithm for numerical function optimization By: Dr. A. H. Kashan

 A sports league is an organization that exists to provide a regulated competition for a number of teams to compete in a specific sport.  Formations are a method of positioning players on the pitch to allow a team to play according to its pre-set tactics.  The main aim of match analysis is:  to identify strengths (S) which can then be further built upon,  to identify weaknesses (W) which suggest areas for improvement,  to use data to try to counter opposing strengths (threats (T)) and exploit weaknesses (opportunities (O))  This kind of analysis is typically known as strengths/weaknesses/opportunities/ threats (SWOT) analysis  The SWOT analysis, explicitly links internal (S/W) and external factors (O/T).  Identification of SWOTs is essential because subsequent steps in the process of planning for achievement of the selected objective may be derived from the SWOTs. 6 League Championship Algorithm: A new algorithm for numerical function optimization By: Dr. A. H. Kashan

 In strategic planning there are four basic categories of matches for which strategic alternatives can be considered: S/T matches show the strengths in light of major threats from competitors. The team should use its strengths to avoid or defuse threats. S/O matches show the strengths and opportunities. Essentially, the team should attempt to use its strengths to exploit opportunities. W/T matches show the weaknesses against existing threats. Essentially, the team must attempt to minimize its weaknesses and avoid threats. These strategy alternatives are generally defensive. W/O matches illustrate the weaknesses coupled with major opportunities. The team should try to overcome its weaknesses by taking advantage of opportunities.  The SWOT analysis provides a structured approach to conduct the gap analysis. A gap is “the space between where we are and where we want to be”.  A transfer is the action taken whenever a player moves between clubs. 7 League Championship Algorithm: A new algorithm for numerical function optimization By: Dr. A. H. Kashan

8 League Championship Algorithm: A new algorithm for numerical function optimization By: A. H. Kashan  LCA, is a population based algorithmic framework for global optimization over a continuous search space.  A common feature among all population based algorithms is that they attempt to move a population of possible solutions to promising areas of the search space, in terms of the problem’s objective, during seeking the optimum.

9 League Championship Algorithm: A new algorithm for numerical function optimization By: A. H. Kashan Sporting terminology (LCA) Leagueweek Team i formationplaying strengthMaximum iterations Evolutionary terminology Populationiteration i th member in the population solution fitness value Number of seasons

10 1)It is more likely that a team with better playing strength wins the game. 2)The outcome of a game is not predictable given known the teams’ playing strength perfectly. It is not unlikely that the world leading FC BARCELONA loses the game to ZORRAT-KARANE-PARS-ABAD from Iranian 3rd soccer division. 3)The probability that team i beats team j is assumed equal from both teams point of view. 4)The outcome of the game is only win or loss (We will later break this rule). 5)Any strength helped team i to win from team j has a dual weakness caused j to lose. In other words, any weakness is a lack of a particular strength. 6)Teams only focus on their upcoming match without regards of the other future matches. Formation settings are done just based on the previous week events. League Championship Algorithm: A new algorithm for numerical function optimization By: A. H. Kashan

11 an n dimensional numerical function that should be minimized over the decision space defined by A formation (a potential solution) for team i at week t indicates the fitness/function value resultant from the best formation for team i experienced till week t  To determine, a greedy selection is done at each iteration as follows: League Championship Algorithm: A new algorithm for numerical function optimization By: A. H. Kashan

is t< S×(L-1) ? Week 1Week 2. Week L-1 Team 1 Team 2 Team L 1. t=1 2. initialize team formations 3. initialize best formations A League schedule is generated 1. Through an artificial match analysis, changes are done in the team formation (new solution) 2. The playing strength along with the resultant formation is determined (fitness calculation) 3. current best formation is updated. Teams play in pairs based on the league schedule at week t, and winner/ loser are determined using a playing strength based criterion; Is it the end of the season ? YES Do possible transfers for each team Terminate NO Week 1Week 2Week L-1 Team 1 Team 2 Team L NO YES t +1  t Start

13 League Championship Algorithm: A new algorithm for numerical function optimization By: A. H. Kashan

 In an ideal league environment we can assume a linear relationship between the team’s playing strength and the outcome of its game.  proportional to its playing strength, each team may have a chance to win (idealized rule 2)  we determine the winner/loser in a stochastic manner by allowing teams to have their chance of win based on their degree of fit  The degree of fit is proportional to the team’s playing strength and is measured based on the distance with an ideal reference point. 14

 We assume that a better team can comply with more factors that an ideal team owns.  Consider teams i and j to fight at week t. Define as the expected chance of team i to beat team j at week t and idealized rule 1 idealized rule 3  Since teams are evaluated based on their distance with a common reference, the ratio of distances determines the winning portions.  A random number in [0,1] is generated, if it is less than or equal to team i wins and team j losses; otherwise j wins and i losses (idealized rule 4). 15 League Championship Algorithm: A new algorithm for numerical function optimization By: A. H. Kashan

l= Index of the team that will play with team i based on the league schedule at week t+1. j= Index of the team that has played with team i based on the league schedule at week t. k= Index of the team that has played with team l based on the league schedule at week t. 16 League Championship Algorithm: A new algorithm for numerical function optimization By: A. H. Kashan

Yes No Could we WIN the game from team j at week t ? Idealized rule 5 the loss is directly due to our WEAKNESSES the success is directly due to the WEAKNESSES of team j the success is directly due to our STRENGTHES the loss is directly due to the STRENGTHES of team j Artificial match analysis doing by team i (S/W evaluation) 17 Idealized rule 5 League Championship Algorithm: A new algorithm for numerical function optimization By: A. H. Kashan

Artificial match analysis doing by team i (O/T evaluation) 18 Could our opponent WIN the game from team k at week t ? No Yes the opponent’s style of play might be a direct THREAT the opponent’s style of play might be a direct OPPORTUNITY Threats are the results of their playing STRENGTHES Opportunities are the results of their playing WEAKNESSES Focusing on the STRENGTHES of team k, gives us a way of affording the possible opportunities Focusing on the WEAKNESSES of team k, gives us a way of avoiding the possible threats Idealized rule 5 League Championship Algorithm: A new algorithm for numerical function optimization By: A. H. Kashan

S/T strategy S/O strategy W/T strategy 19 W/O strategy League Championship Algorithm: A new algorithm for numerical function optimization By: A. H. Kashan

 Assume that team k has won the game from team l. To beat l, it is reasonable that team i devises a playing style rather similar to that was adopted by team k at week t.  By “ ” we address the gap between the playing style of team i and team k, sensed via “focusing on the strengths of team k”.  In a similar way we can interpret “ ” when “focusing on the weaknesses of team k”.  In other words, it may be reasonable to avoid a playing style rather similar to that was adopted by team k.  We can interpret “ ” or “ ” in a similar manner. 20 League Championship Algorithm: A new algorithm for numerical function optimization By: A. H. Kashan

21 If i was winner and l was winner, then (S/T equation): Else if i was winner and l was loser, then (S/O equation): Else if i was loser and l was winner, then (W/T equation): Else if i was loser and l was loser, then the (W/O equation): End if League Championship Algorithm: A new algorithm for numerical function optimization By: A. H. Kashan

 In above formulas we rely upon the fact that normally teams play based on their current best formation (that found it suitable over the times), while preparing the required changes recommended by the match analysis.  and are constant coefficients used to scale the contribution of “retreat” or “approach” components, respectively.  the diversification is controlled by allowing to “retreat” from a solution and also by coefficient, while the intensification is implicitly controlled by getting “approach” to a solution and by coefficient.  We refer the above system of updating equations as LCA/recent since they use the teams’ most recent formation as a basis to determine the new formations. 22 League Championship Algorithm: A new algorithm for numerical function optimization By: A. H. Kashan

23 If i was winner and l was winner, then (S/T equation): Else if i was winner and l was loser, then (S/O equation): Else if i was loser and l was winner, then (W/T equation): Else if i was loser and l was loser, then the (W/O equation): End if League Championship Algorithm: A new algorithm for numerical function optimization By: A. H. Kashan

24  It is unusual that coaches do changes in all or many aspects of the team. normally a few number of changes are devised.  To simulate the number of changes ( ) made in, we use a truncated geometric distribution.  Where r is a random number in [0,1] and is a control parameter. is the least number of changes realized during the artificial match analysis  number of dimensions are selected randomly from and their value is changed according to one of the Equations League Championship Algorithm: A new algorithm for numerical function optimization By: A. H. Kashan

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26  Comparison is done between LCA and the highly recognized (PSO) algorithm League Championship Algorithm: A new algorithm for numerical function optimization By: A. H. Kashan

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 In order to see that whether each of S/T, S/O, W/T and W/O updating equations has a significant effect on the performance of LCA, we sequentially omit the possible effect that each equation might have on the evolution of the solutions. 31

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Learning from team’s previous game only If i was winner, then (S equation): Else if i was loser, then (W equation): End if Learning from opponent’s previous game only If l was winner, then (T equation): Else if l was loser, then (O equation): End if 33

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Interestingly, these empirical results are in accordance with the business reality. In business strategy there are two schools of thought, the “environmental (external)” and the “resource based (internal)”. Through 1970s and 80s, the dominant school was the environmental school which dictates that a firm should analyze the forces present within the environment in order to asses the profit potential of the industry. Nevertheless, above average performance is more likely to be the result of core capabilities inherent in a firm’s resources (internal view) than its competitive positioning in its industry (external view). 35

Tie outcome is interpreted as the consequent of the strengths/ opportunities and weaknesses/threats 36

Tie outcome is neutral. There is no learning from ties 37

Tie outcome is randomly interpreted as win or loss For example, in this situation, under the case of “Else if i was winner and l had tied” the new formation is set up as follows: Tie outcome is interpreted as win If i had won/tied and l had won/tied, then use (S/T) equation to setup a new formation Else if i had won/tied and l was loser, then use (S/O) equation setup a new formation Else if i was loser and l had won/tied, then use (W/T) equation to setup a new formation Else if i was loser and l was loser, then use (W/O) equation to setup a new formation End if 38

Tie outcome is interpreted as loss If i was winner and l was winner, then use (S/T) equation to setup a new formation Else if i was winner and l had lost/tied, then use (S/O) equation setup a new formation Else if i had lost/tied and l was winner, then use (W/T) equation to setup a new formation Else if i had lost/tied and l had lost/tied, then use (W/O) equation to setup a new formation End if 39

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