Study the problem of minimizing the total number of service stations or the construction cost on the premise of meeting the requirements of customers at all demand points
The set coverage problem studies the problem of minimizing the total number of service stations or the construction cost on the premise that all customers at demand points are covered.Set coverage problemIt was first proposed by Roth and Toregas to solve the problems of fire control center andambulanceFor the location problem of emergency service facilities, they respectively established an integer programming model for the set coverage problem under the conditions of different service station construction costs and the same cost.
Coverage problems can be divided into two categories: Maximum Covering Location Problem (MCLP) and Location Set Covering Problem (LSCP).[1-2]
Detailed introduction
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Set coverage problem
Later, Minieka, Moore and ReVelle continued their researchSet coverage problem。Plane and Hendrick, Daskin and Stern established a double objective set coverage problem with the minimum number of service stations and the maximum number of customers covered by backup.Heung Suk Huang studied the dynamic set coverage problem when the product will deteriorate or improve over time.In recent decades, many heuristic algorithms have been used to solve the set coverage problem. M.L. Fisher and P. Kedia proposed a heuristic algorithm based on duality to solve the set coverage problem with up to 200 candidate points and 2000 demand points;Beasley J.E. and JornstenK combines subgradient optimization method with Lagrangian relaxation algorithm to solve such problems;Marcos Alminana and Jesus T. Pastor applied the agent heuristic algorithm to solve the set coverage problem.J. E. Beasley and P.C. Chu proposed a genetic algorithm to solve the coverage problem of different time sets of service station construction costs.Grossman and Wool [56] used a lot of experiments to compare nine heuristic algorithms for solving SCLP. Among them, the random greedy algorithm (R-Gr), the simple greedy algorithm (S-Gr), and the transformed greedy algorithm (Alt Gr) are among the first four best algorithms in almost all problems. Among them, the random greedy algorithm performs best, obtaining the best solution 56 times out of 60 random problems.Karp proved that the set covering problem is NP complete.
Maximum coverage problem
The maximum coverage problem or P-coverage problem is to study how to set up P service stations to maximize the demand for acceptable services when the number of service stations and service radius are known.Like other basic problems, the maximum network coverage problem is also an NP hard problem (Marks. Daskin).The initial maximum coverage problem was proposed by Church RL and ReVelle C. They limited the optimal location of service stations to network nodes;Church RL and Meadows ME have given the optimal algorithm in general in the determined set of key candidate nodes. They solve it by linear programming. If the optimal solution is not an integer, they use the branch and bound method to solve it;Church and Meadows proposed the pseudo Hakimi property of the maximum coverage problem, that is, in any network, there is an extended set of finite nodes, in which at least one optimal solution of the maximum coverage problem is included.Benedict, Hogan, ReVelle, and Daskin consider the maximum coverage problem in the case of congested service systems. They regard the probability of any service station being busy as an exogenous variable, and the objective function is to maximize the expected demand that the service station can cover.Haldun Aytug and Cem Saydam used genetic algorithms to solve large-scale maximum expected coverage problems, and made a comparison.Fernando Y et al. compared the queuing and non queuing situations in the maximum expected coverage problem.Berman studied the relationship between the maximum covering problem and the partial covering problem.Oded Berman and DmitryKrass, Oded Berman, Dmitry Krass and Zvi Drezner discuss the maximum cover problem which is more general than the traditional maximum cover problem, and give a Lagrangian relaxation algorithm.Orhan Karasakal and Esra K. Karasakal discussed partial coverage and defined the coverage degree.Jorge H. Jaramillo, Joy Bhadury and Rajan Batta in thegenetic algorithmThe operation strategy of genetic algorithm for maximum cover problem is introduced in application research.
