The presence of several items leads then to a variation in the overall utility computation, which will be called synergy: therefore, the robustness of the problem co-exists with the polynomial utility terms. Among the several different interpretations given for the aforementioned problems, for this paper it was chosen to deal with a set of investments as ‘items’, whose cost can vary in a bounded interval. Given that the robust and the polynomial KP formulations have been always treated separately, the goal of this paper is to mix the two problems to fill such gap in the literature. If on the one hand the RKP more dynamically models the Knapsack Problem, on the other hand its framework does not originally include more than the singular items’ contributions. Intuitively, the robust solution considers the worst case scenario, occurring when all the selected items achieve their respective largest allowed weight. The maximum number of items whose cost varies is governed by a parameter Γ, which is supposed to have been statistically estimated. It represents an extension of the KP where the weight of each item varies in a pre-defined interval. The Robust Knapsack Problem (RKP) is one possible way to face this lack of information. However, in several settings this may not be the case, as uncertainty is present over some of the items’ characteristics. its profit and cost) is indeed known a priori. The framework inside which the two aforementioned problems are defined only deals with a deterministic scenario: the informative content about each item (i.e. To the present knowledge, and probably due to this enhanced complexity, it has never been specifically studied in the literature. In order to linearize the products of the binary variables of each degree, this problem needs an exponential number of variables and constraints in the real setting, the decision maker cannot define all such contributions, but in this way a more flexible model is provided, so that a limited amount of multiple contributions brought by increasing numbers of items can be taken into account if necessary. The Polynomial Knapsack Problem (PKP) is a further generalization of the Quadratic Knapsack Problem up to any polynomial degree terms. Indeed, given a set of items, each with a weight and a value, an extra-profit is considered if two items are selected in the solution. In order to fill this gap, the Quadratic Knapsack Problem (QKP) was introduced, extending the problem formulation to model the impact of quadratic terms in the utility function. Indeed, it only considers a utility reward measure from the singular item, without modelling dependencies of contributions common to several items. However, the basic formulation of the KP is limiting with respect to real application scenarios. The problem often arises in resource allocation, where the decision maker has to choose from a set of non-divisible projects or tasks under a fixed budget. The KP is a well known combinatorial optimization problem which given a set of items, each one characterized by a weight and associated to a utility value, aims at selecting the subset of items maximizing the total utility, by respecting a budget constraint on the items’ weight. We call this new problem Polynomial Robust Knapsack Problem (PRKP). This work aims at correctly mapping these practical situations into a suitable formulation, representing the joint development of two sub-classes of the traditional Knapsack Problem (KP), namely the Robust Knapsack Problem and the Polynomial Knapsack Problem. the common activation of a metro station and a parking area nearby might produce a traffic decrease bigger than the sum of those estimated for each single building). An effective example is when the decision maker is asked to choose among many conflicting projects, which may create some further extra economic value in the form of synergies among them (e.g. More complicated decisions require, instead, a more systematic approach, and the adoption of proper methodologies of decision support, especially if such complex problems require to take into consideration a plurality of points of view. Our lives are the result of decisions most of them are taken on the basis of intuition and common sense.
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