A Survey of League Championship Algorithm: Prospects and Challenges <s> Subset Problems <s> The Maximum Independent Set (MIS) problem is a well-known problem where the aim is to find the maximum cardinality independent set in an associated graph. Map labeling problems can often be modeled as a MIS problem in a conflict graph, where labels are selected to be placed near graphical features not allowing overlaps (conflicts) between labels or between labels and features. However, the MIS problem is NP-hard and exact techniques present difficulties for solving some instances. Thus, this paper presents a Lagrangean decomposition to solve a map labeling problem, the Point-Feature Label Placement problem. We treated the problem by a conflict graph that is partitioned into small sub-problems and copies of some decision variables are done. These copies are used in the sub-problems constraints and in some constraints to ensure the equality between the original variables and their copies. After these steps, we relax these copy constraints in a Lagrangean way. Using instances proposed in the literature, our approach was able to prove the optimality for all of them, except one, and the results were better than the ones provided by a commercial solver. <s> BIB001 </s> A Survey of League Championship Algorithm: Prospects and Challenges <s> Subset Problems <s> Given a graph G=(V,E), the independent set problem is that of finding a maximum-cardinality subset S of V such that no two vertices in S are adjacent. We introduce two fast local search routines for this problem. The first can determine in linear time whether a maximal solution can be improved by replacing a single vertex with two others. The second routine can determine in O(mΔ) time (where Δ is the highest degree in the graph) whether there are two solution vertices than can be replaced by a set of three. We also present a more elaborate heuristic that successfully applies local search to find near-optimum solutions to a wide variety of instances. We test our algorithms on instances from the literature as well as on new ones proposed in this paper. <s> BIB002 </s> A Survey of League Championship Algorithm: Prospects and Challenges <s> Subset Problems <s> This paper considers the multi-objective reliability redundancy allocation problem of a series system where the reliability of the system and the corresponding designing cost are considered as two different objectives. Due to non-stochastic uncertain and conflicting factors it is difficult to reduce the cost of the system and improve the reliability of the system simultaneously. In such situations, the decision making is difficult, and the presence of multi-objectives gives rise to multi-objective optimization problem (MOOP), which leads to Pareto optimal solutions instead of a single optimal solution. However in order to make the model more flexible and adaptable to human decision process, the optimization model can be expressed as fuzzy nonlinear programming problems with fuzzy numbers. Thus in a fuzzy environment, a fuzzy multi-objective optimization problem (FMOOP) is formulated from the original crisp optimization problem. In order to solve the resultant problem, a crisp optimization problem is reformulated from FMOOP by taking into account the preference of decision maker regarding cost and reliability goals and then particle swarm optimization is applied to solve the resulting fuzzified MOOP under a number of constraints. The approach has been demonstrated through the case study of a pharmaceutical plant situated in the northern part of India. <s> BIB003 </s> A Survey of League Championship Algorithm: Prospects and Challenges <s> Subset Problems <s> Composite beams, plates and shells are widely used in the aerospace industry because of their advantages over the commonly used isotropic structures especially when it comes to weight savings. Buckling analyses of composite structural components must be performed in order to ensure, for instance, that a composite panel designed to be part of a control surface does not buckle thereby compromising its aerodynamic shape. Optimization of composite structures has been performed in this paper using Genetic algorithm. Genetic algorithm (GA) approaches are successfully implemented for the TSP. The buckling load of composite plate, which is obtained by the Artificial Neural Networks, was used as the fitness function in the GA to find its optimized value by arranging the ply stacking sequence. <s> BIB004 </s> A Survey of League Championship Algorithm: Prospects and Challenges <s> Subset Problems <s> The Quadratic Knapsack Problem (QKP) is one of the well-known combinatorial optimization problems. If more than one knapsack exists, then the problem is called a Quadratic Multiple Knapsack Problem (QMKP). Recently, knapsack problems with setups have been considered in the literature. In these studies, when an item is assigned to a knapsack, its setup cost for the class also has to be accounted for in the knapsack. In this study, the QMKP with setups is generalized taking into account the setup constraint, assignment conditions and the knapsack preferences of the items. The developed model is called Generalized Quadratic Multiple Knapsack Problem (G-QMKP). Since the G-QMKP is an NP-hard problem, two different meta-heuristic solution approaches are offered for solving the G-QMKP. The first is a genetic algorithm (GA), and the second is a hybrid solution approach which combines a feasible value based modified subgradient (F-MSG) algorithm and GA. The performances of the proposed solution approaches are shown by using randomly generated test instances. In addition, a case study is realized in a plastic injection molding manufacturing company. It is shown that the proposed hybrid solution approach can be successfully used for assigning jobs to machines in production with plastic injection, and good solutions can be obtained in a reasonable time for a large scale real-life problem. <s> BIB005 </s> A Survey of League Championship Algorithm: Prospects and Challenges <s> Subset Problems <s> Iterated greedy search is a simple and effective metaheuristic for combinatorial problems. Its flexibility enables the incorporation of components from other metaheuristics with the aim of obtaining effective and powerful hybrid approaches. We propose a tabu-enhanced destruction mechanism for iterated greedy search that records the last removed objects and avoids removing them again in subsequent iterations. The aim is to provide a more diversified and successful search process with regards to the standard destruction mechanism, which selects the solution components for removal completely at random. <s> BIB006 </s> A Survey of League Championship Algorithm: Prospects and Challenges <s> Subset Problems <s> The maximum independent set Problem is to find a biggest verte x independent set in a given undirected graph. It is a vitally important NP problem in graph theory and applied mathematics, having numerous real life applications. It can be difficul tly solved by the electronic computer in exponential level time. Simultaneity in previous studies DNA molecular computation usually be used to solve NP-complete continuous path search problems (for example HPP, traveling salesman problem), rarely for NP problems with discrete vertex or path solutions result, such as the maximum independent set problem, graph coloring problem and so on. In this paper, we present a new algorithm for solving the maximum independent set problem with DNA molecular operations. For an undirected graph with n vertices, We reasonably design fixed length DNA strands repr esenting the vertices and edges of graph, take appropriate steps and get the solutions of the problem in proper length range using O(n 2 ) time. We extend the application of DNA molecular operations and simultaneity simplify the complexity of the computation. <s> BIB007 </s> A Survey of League Championship Algorithm: Prospects and Challenges <s> Subset Problems <s> This work presents a novel orthogonal simplified swarm optimization scheme (OSSO) that combines repetitive orthogonal array testing (ROA), re-initialize population (RIP), and SSO for solving intractable large-scale engineering problems. This scheme is applied to the series–parallel redundancy allocation problem (RAP) with a mix of components. RAP involves setting reliability objectives for components or subsystems in order to meet the resource consumption constraint, e.g., the total cost. RAP has been an active area of research for the past four decades. The difficulties confronted by RAP are to maintain feasibility with respect to three nonlinear constraints, namely, cost-, weight-, and volume-related constraints. As evidence of the utility of the proposed approach, we present extensive computational results on random test problems. The computational results compare favorably with previously developed algorithms in the literature. The results in this paper show that the proposed OSSO can perform excellently in a limited computation time. <s> BIB008 </s> A Survey of League Championship Algorithm: Prospects and Challenges <s> Subset Problems <s> In this paper, an open-shop scheduling problem with stochastic process times is considered. The objective function is to minimize the weighted sum of earliness and tardiness costs. This type of objective function, in addition to the optimal schedule, puts forward the issue of finding optimal values of start times of jobs. In scheduling problems with stochastic process times, unlike the deterministic variants, it is not possible to determine the schedule before the random variables are realized. Whenever a machine becomes available (for example finishes processing some jobs) a job should be selected to be passed on it from the set of available jobs. This selection process is carried out in real-time. We have devised a simulation-optimization algorithm which calculates the distribution functions of the completion times of jobs via central limit theorem. These distribution functions are used for cost estimation in the real-time job selecting process. In other words, whenever some jobs are competing for a free machine, a pair wise cost competition is raised among them and the winner is passed on the free machine. Moreover, the distribution function of the completion time of each job is also used to adjust its start time. Finally, some test problems are generated and solved by our algorithm. The results demonstrate that our algorithm, even without adjusting the start times, outperforms the algorithm which acts randomly in the real-time job selecting process. <s> BIB009
Multiple knapsack problem  BIB005 BIB006 , maximum independent set problem BIB002 BIB001 BIB007 , maximum clique problem   and redundancy allocation problem BIB003 BIB008 , etc. Other Applications Open shop BIB009 and group shop scheduling across researches that only relay a algorithm developed at a certain number of generations. But it is not normally clear, if such an algorithm is the outcome of a single run or the mean of many runs. More so, in some cases it is not very clear how to choose when to discontinue getting a good system. It is also uncertain how fit the LCA might generalize to diverse situations in these cases BIB004 . Most of the related LCA literatures show that some enhancements need to be done to suite a particular situation. Different problems have different conditions and parameters, and therefore may need different approach to solve. This problem of generalization is not particular to the LCA as most optimization and heuristic algorithms also suffered from this issue. In essence, more comprehensive studies of generalization and standardization in LCA will greatly improve it applicability in other areas.