3 objective problem test cases

A set of randomly generated 3 objective knapsack, assignment and travelling salesman problems used to test optimisation algorithms.<br><br>These are generated using similar techniques to those of Laumanns et al. (2006) (for knapsack problems), Przybylski et al. (2010) (for assignment problems), and Özpeynirci and Köksalan (2010) (for traveling salesman problems). A knapsack instance is generated by randomly assigned an integer weight (uniformly at random in the range {60, ... , 100}) to each of n items. The upper bound on the total weight of the selected items is set to be half of the total weight of all items. Each objective function is chosen in a similar manner, with the coefficients for each item drawn uniformly<br>at random from the range {[60, ... , 100}.<br><br>Assignment problems are generated in the manner of Przybylski et al. (2010), with objective function coefficients drawn uniformly at random from {0, ... , 20}. We also generate instances of the traveling salesman problem as per Özpeynirci and Köksalan (2010). We place cities on a 1000 × 1000 plane by assigning integer coordinates to cities, and round the Euclidean distance between any two cities to an integer value.<br><br>Finally some non-Euclidean traveling salesman problems are generated by assigning distances between towns uniformly at random from {20, ..., 180}. Note that these distances will not satisfy common properties of distances or norms (such as the triangle inequality).<br><br>Laumanns M, Thiele L, Zitzler E (2006) An efficient, adaptive parameter variation scheme for metaheuristics based on the epsilon-constraint method. European Journal of Operational Research 169(3):932 – 942, ISSN 0377-2217, URL http://dx.doi.org/http://dx.doi.org/10.1016/j.ejor.2004.08.029.<br><br>Przybylski A, Gandibleux X, Ehrgott M (2010) A two phase method for multi-objective integer programming and its application to the assignment problem with three objectives. Discrete Optimization 7(3):149 – 165, ISSN 1572-5286, URL http://dx.doi.org/https://doi.org/10.1016/j.disopt.2010.03.005.<br><br>Özpeynirci O, Köksalan M (2010) An exact algorithm for finding extreme supported nondominated points of multiobjective mixed integer programs. Management Science 56(12):2302–2315, URL http://dx.doi.org/10.1287/mnsc.1100.1248.<br><br>These use an extended LP file format where multiple objectives are defined as additional constraints after the original problem's constraints. The right-hand-side value of the last constraint defines the number of objectives.本数据集包含一组随机生成的三目标背包问题（knapsack problem）、指派问题（assignment problem）与旅行商问题（travelling salesman problem），用于优化算法的性能测试。 三类问题的生成方法分别参考了Laumanns等人（2006）（针对背包问题）、Przybylski等人（2010）（针对指派问题）以及Özpeynirci与Köksalan（2010）（针对旅行商问题）的相关技术。其中，背包问题实例的生成方式为：为n个物品分别随机分配整数重量（在{60, ..., 100}区间内均匀随机取值），将选中物品的总重量上限设置为所有物品总重量的一半；每个目标函数的生成方式与之类似，每个物品的系数从{60, ..., 100}区间内均匀随机抽取。 指派问题的生成遵循Przybylski等人（2010）的方法，其目标函数系数从{0, ..., 20}区间内均匀随机抽取。旅行商问题的生成则参考Özpeynirci与Köksalan（2010）的方案：将城市坐标设置为1000×1000平面内的整数坐标，并将任意两座城市间的欧氏距离取整为整数值。 此外本数据集还包含部分非欧旅行商问题：城镇间的距离从{20, ..., 180}区间内均匀随机生成。需注意，此类距离不满足距离或范数的通用性质（如三角不等式）。 Laumanns M, Thiele L, Zitzler E (2006) 一种基于ε约束法的元启发式算法高效自适应参数调整方案. 欧洲运筹学杂志, 169(3):932–942, ISSN 0377-2217, 链接: http://dx.doi.org/http://dx.doi.org/10.1016/j.ejor.2004.08.029. Przybylski A, Gandibleux X, Ehrgott M (2010) 一种求解多目标整数规划的两阶段方法及其在三目标指派问题中的应用. 离散优化, 7(3):149–165, ISSN 1572-5286, 链接: http://dx.doi.org/https://doi.org/10.1016/j.disopt.2010.03.005. Özpeynirci O, Köksalan M (2010) 一种求解多目标混合整数规划极端支撑非支配点的精确算法. 管理科学, 56(12):2302–2315, 链接: http://dx.doi.org/10.1287/mnsc.1100.1248. 本数据集采用扩展LP文件格式，其中多目标以原始问题约束之后的附加约束形式定义，且最后一条约束的右侧值即为目标函数的数量。