The League Championship Algorithm: A new algorithm for numerical function optimization

1. The League Championship Algorithm: • A new algorithm for numerical • function optimization • By: A. H. Kashani

2. Introduction • 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. League Championship Algorithm: A new algorithm for numerical function optimization By: Dr. A. H. Kashan

3. Someexamples • 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. League Championship Algorithm: A new algorithm for numerical function optimization By: Dr. A. H. Kashan

4. 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 Are inspired by nature’s capability to evolve living beings well adapted to their environment Metaheuristics Evolutionary algorithms Trajectory methods Swarm intelligence Social, political, music, sport , etc • Tabu search • Variable neighborhood search • Harmony search • Society and civilization • Imperialist competitive algorithm • League championship • algorithm • Evolution strategies • Genetic programming • Genetic algorithm • Ant colony optimization • Particle swarm optimization • Artificial bee colony • Bacterial foraging • optimization • Group search optimizer League Championship Algorithm: A new algorithm for numerical function optimization By: Dr. A. H. Kashan

5. The league championship algorithm (LCA)

6. A review on the sporting terminology and required background • A sportsleagueis 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 matchanalysisis: • 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. League Championship Algorithm: A new algorithm for numerical function optimization By: Dr. A. H. Kashan

7. A review on the sporting terminology and required background • 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 gapanalysis. A gap is “thespace between where we are and where we want to be”. • Atransferis the action taken whenever a player moves between clubs. League Championship Algorithm: A new algorithm for numerical function optimization By: Dr. A. H. Kashan

8. LCA as an EA • 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. League Championship Algorithm: A new algorithm for numerical function optimization By: A. H. Kashan

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

10. Idealized rules • It is more likely that a team with better playing strength wins the game. • 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. • The probability that team i beats team j is assumed equal from both teams point of view. • The outcome of the game is only win or loss (We will later break this rule). • 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. • 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. Notations • 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

12. A League schedule is generated t +1t Teams play in pairs based on the league schedule at week t, and winner/ loser are determined using a playing strength based criterion; Terminate YES Is it the end of the season? 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. NO NO 1. t=1 2. initialize team formations 3. initialize best formations YES is t< S×(L-1) ? Do possible transfers for each team Start

13. Generating the league schedule 1 2 3 4 1 2 3 4 1 5 2 3 5 6 7 8 5 6 7 8 6 7 8 4 week 1 week 2 week 3 1 3 4 8 1 6 5 2 2 5 6 7 7 8 4 3 week 7 week 4 League Championship Algorithm: A new algorithm for numerical function optimization By: A. H. Kashan

14. Determining winner/loser • 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 (idealizedrule2) • 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.

15. Determining winner/loser • 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). League Championship Algorithm: A new algorithm for numerical function optimization By: A. H. Kashan

16. Setting up a new formation for team i 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. League Championship Algorithm: A new algorithm for numerical function optimization By: A. H. Kashan

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

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

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

20. Setting up a new team formation • 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. Setting up a new team formation Ifi was winner and l was winner, then (S/T equation): Elseifi was winner and l was loser, then (S/O equation): Elseifi was loser and l was winner, then (W/T equation): Elseifi was loser and l was loser, then the (W/O equation): Endif 21 League Championship Algorithm: A new algorithm for numerical function optimization By: A. H. Kashan

22. Setting up a new team formation • Inabove 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. League Championship Algorithm: A new algorithm for numerical function optimization By: A. H. Kashan

23. LCA/best: A variant Ifi was winner and l was winner, then (S/T equation): Elseifi was winner and l was loser, then (S/O equation): Elseifi was loser and l was winner, then (W/T equation): Elseifi was loser and l was loser, then the (W/O equation): Endif 23 League Championship Algorithm: A new algorithm for numerical function optimization By: A. H. Kashan

24. How big would be the number of changes? • 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 24 League Championship Algorithm: A new algorithm for numerical function optimization By: A. H. Kashan

25. Test functions

26. Parameter settings • 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

27. Comparison study

28. Comparison study 28

29. Visualization on Six-Hump Camelback function Week 1 Week 5 Week 10 Week 20

30. Visualization on Six-Hump Camelback function Week 50 Week 100

31. Effect of LCA updating equations • 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.

32. Effect of LCA updating equations

33. Effect of adopting different learning strategies in the artificial post-match analysis Learning from team’s previous game only Ifi was winner, then (S equation): Else if i was loser, then (W equation): Endif Learning from opponent’s previous game only If l was winner, then (T equation): Else if l was loser, then (O equation): Endif

34. Effect of adopting different learning strategies in the artificial post-match analysis

35. Effect of adopting different learning strategies in the artificial post-match analysis • 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 (internalview) than its competitive positioning in its industry (externalview).

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

37. Inclusion of the tie outcome Tie outcome is neutral. There is no learning from ties

38. Inclusion of the tie outcome 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 Ifi had won/tied and l had won/tied, then use (S/T) equation to setup a new formation Elseifi had won/tied and l was loser, then use (S/O) equation setup a new formation Elseifi was loser and l had won/tied, then use (W/T) equation to setup a new formation Elseifi was loser and l was loser, then use (W/O) equation to setup a new formation End if

39. Inclusion of the tie outcome Tie outcome is interpreted as loss Ifi was winner and l was winner, then use (S/T) equation to setup a new formation Elseifi was winner and l had lost/tied, then use (S/O) equation setup a new formation Elseifi had lost/tied and l was winner, then use (W/T) equation to setup a new formation Elseifi had lost/tied and l had lost/tied, then use (W/O) equation to setup a new formation End if

40. Inclusion of the tie outcome

41. Inclusion of the end season transfers

42. Inclusion of the end season transfers

43. Thanks for your attention!