Binary Knapsack Problem (BKP) is to select a subset of an element (item) set with the highest value while keeping the total weight within the capacity of the knapsack. is paper presents an integer programming model for a variation of BKP where the value of each element may depend on selecting or ignoring other elements. Strengths of such Value-Related Dependencies are assumed to be imprecise and hard to specify. To capture this imprecision, we have proposed modeling value-related dependencies using fuzzy graphs and their algebraic structure.
KEYWORDSBinary Knapsack Problem, Integer Programming, Dependency, Value
MODELING DEPENDENCIESFuzzy graphs have demonstrated to properly capture imprecision of real world problems [4,6]. Hence, we have used algebraic structure of fuzzy graphs for capturing the imprecision associated with strengths of value-related dependencies. We have specially modied the classical de nition of fuzzy graphs in order to consider not only the strengths but also the qualities (positive or negative) [5-7] of value-related dependencies (De nition 1).De nition 1. Value Dependency Graph (VDG). A VDG is a signed directed fuzzy graph [11] G = (E, σ, ρ) in which a non-empty set of elements E : {e 1 , ..., e n } constitute the graph nodes. Also, the qualitative function σ : E × E → {+, −, ±} and the membership function ρ : E×E → [0, 1] denote qualities and strengths of explicit value-related dependencies receptively. As such, a pair of elements (e i , e j ) with ρ i, j 0 and σ i, j ± denotes an explicit value-related dependency from e i to e j . It is clear that we have ρ i, j = 0 if the value of an element e i is not explicitly in uenced by selecting or ignoring e j . In such cases we have σ i, j = ± where ± denotes the quality of (e i , e j ) is non-speci ed.De nition 2. Value-Related Dependencies. A value-related dependency in a VDG G = (E, σ, ρ) is de ned as a sequence of elements d i : e(1), ..., e(k) such that for each e(j) in d i , 2 ≤ j ≤ k, we have ρ j−1, j 0. A consecutive pair e(j − 1), e(j) speci es an explicit value-related dependency from e(j − 1) to e(j).