Parametric Solution for Linear Bicriteria Knapsack Models
针对双准则线性加权背包问题，提出一种基于网络最长路径的参数化解算法，能在变量数与资源限制的乘积时间内给出任意权重组合下的最优解，并通过计算实验验证了效率。
Linear weighing is a common approach to handle multiple criteria and the “knapsack” is a well-known combinatorial optimization problem. A knapsack problem with two linearly weighted, objective criteria is considered in this paper. For better support, it is important to provide the decision maker with information that covers the whole range of alternatives. Toward this goal, an algorithm for the construction of a parametric solution to the problem, i.e., for any combination of weights, is developed, which is based on finding a longest path in a network which compactly represents all feasible solutions to the knapsack problem. Exploiting the special structure of the knapsack model, the algorithm efficiently constructs the parametric solution in time that is linear in the product of the number of variables, the resource limit (right-hand side of the constraint), and the (finite) number of vectors which constitute the solution. The amount of memory required is linear in the product of the number of variables and the resource limit. Results of computational study are reported. The results are used to assess the efficiency of the algorithm and characterize its behavior with respect to the parameter values.