The multidimensional 0–1 knapsack problem: An overview

The problem of maximizing diversity deals with selecting a set of elements from some larger collection such that the selected elements exhibit the greatest variety of characteristics. A new model is proposed in which the concept of diversity is quantifiable and measurable. A quadratic zero-one model is formulated for diversity maximization. Based upon the formulation, it is shown that the maximum diversity problem is NP-hard. Two equivalent linear integer programs are then presented that offer progressively greater computational efficiency. Another formulation is also introduced which involves a different diversity objective. An example is given to illustrate how additional considerations can be incorporated into the maximum diversity model.

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Analyzing and Modeling the Maximum Diversity Problem by Zero‐One Programming*

There are numerous situations in management and elsewhere in which an individual decision maker chooses subsets of multiattributed items. The specification of a measure of goodness for selecting subsets may differ from one situation to the next. In this paper, a model is developed for evaluating subsets where the choice criterion is one of balance among the attributes of items in the subset chosen. A method for determining the parameters of the model from a small number of judgments on subsets using linear programming is discussed. The model is applied to the problem of evaluating subsets of television shows and of choosing the most balanced subset of shows. Several extensions of the model and potential applications are also given.

A Balance Model for Evaluating Subsets of Multiattributed Items

The multidimensional knapsack problem (MKP) is one of the widely known integer programming problems. The MKP has received significant attention from the operational research community for its large number of applications. Solving this NP-hard problem remains a very interesting challenge, especially when the number of constraints increases. In this paper we present a k-means transition ranking (KMTR) framework to solve the MKP. This framework has the property to binarize continuous population-based metaheuristics using a data mining k-means technique. In particular we binarize a Cuckoo Search and Black Hole metaheuristics. These techniques were chosen by the difference between their iteration mechanisms. We provide necessary experiments to investigate the role of key ingredients of the framework. Finally to demonstrate the efficiency of our proposal, MKP benchmark instances of the literature show that KMTR competes with the state-of-the-art algorithms.

Claude McMillan

Papers

Discrete optimization: integer programming and network analysis for management decisions

Education.: resource allocation decisions via 0, 1 programming