1. What contributions have the authors mentioned in the paper "Integer programming formulations for three sequential discrete competitive location problems with foresight" ?
The environment of these problems consists of an open market with two non-cooperative firms ( leader and follower ), several customers and locations where facilities can be located.. In order to capture the demand of the customers, the leader enters the market by locating a set of facilities knowing the potential locations where the follower can locate her facilities after the leader ’ s decision.. The authors consider here three pairs of objective functions for the leader/follower previously studied in the literature: maximizing/minimizing the demand captured by the leader, minimizing/maximizing the regret of the leader, maximizing the demand Email addresses: josegentile @ terra.. For each model, the authors propose an linear integer programming formulation with a polynomial number of variables and an exponential number of constraints.. The authors report extensive computational experiments realized on instances inspired by those from the literature, comparing their algorithms with the exact and heuristic algorithms previously published for these problems.
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2. What techniques were used to speed up the process?
Some techniques to speed up the process were used, such as quickly discarding the budgets solutions and follower strategies which produce greater regret than the minimum obtained so far.
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3. What are the constraints for the WC-CMCLP?
Given a relaxed leader solution (xl, yl, zl) ∈ [0, 1]|I| × [0, 1]|I|×|J | × [0, 1]|J |for the model (42) – (58) possibly violating some of the constraints (57), the separation problem amounts to find the follower strategy that maximizes the violation of the constraint, which is to maximize the right side of (57).max ∑ j∈J ∑ k∈F wj ∑ i∈L′(j)|dkj≤dij ylij yfkj (61) s.t. yfkj ≤ xf k , ∀j ∈ J,∀k ∈ F (62)∑k∈Fyfkj = 1, ∀j ∈ J (63)yfkj ∈ {0, 1} , ∀k ∈ F, ∀j ∈ J (64) xf ∈ X f . (65)The exact algorithms for the WC-CMCLP, the RE-CMCLP and the STCMCLP are built on the top of the branch-and-cut algorithms that solve the models given by, respectively, (7) – (12), (19) – (25) and (42) – (58) using commercial solvers.
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4. How do the authors improve the algorithms performance?
For future papers, the authors intend to improve the algorithms performance through two ways: proposing metaheuristics and finding cuts for both the original and the separation problems.
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