An Integer Linear Programming Model for Binary Knapsack Problem with Dependent Item Values

Binary Knapsack Problem (BKP) is to select a subset of items with the highest value while keeping the size within the capacity of the knapsack. This paper presents an Integer Linear Programming (ILP) model for a variation of BKP where the value of an item may depend on presence or absence of other items in the knapsack. 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. We have demonstrated through simulations that our proposed ILP model is scalable to large number of items.

[1]  R. Carraway,et al.  An algorithm for maximizing target achievement in the stochastic knapsack problem with normal returns , 1993 .

[2]  E. Steinberg,et al.  A Preference Order Dynamic Program for a Knapsack Problem with Stochastic Rewards , 1979 .

[3]  Stephen P. Boyd,et al.  Convex Optimization , 2004, Algorithms and Theory of Computation Handbook.

[4]  Andrew Kusiak,et al.  Dependency analysis in constraint negotiation , 1995, IEEE Trans. Syst. Man Cybern..

[5]  W. Sharpe.  Portfolio Theory and Capital Markets , 1970 .

[6]  Hannu Kivijärvi,et al.  Operations Research and Its Practice in Finland , 1986 .

[7]  Davoud Mougouei,et al.  A Fuzzy-Based Technique for Describing Security Requirements of Intrusion Tolerant Systems , 2013 .

[8]  J. Sprenger.  Foundations of a Probabilistic Theory of Causal Strength , 2018, The Philosophical Review.

[9]  Harry M. Markowitz,et al.  Foundations of Portfolio Theory , 1991 .

[10]  Johan de Kleer,et al.  A Qualitative Physics Based on Confluences , 1984, Artif. Intell..

[11]  Michael P. Wellman.  Formulation of tradeoffs in planning under uncertainty , 1988 .

[12]  Mordechai I. Henig.  Risk Criteria in a Stochastic Knapsack Problem , 1990, Oper. Res..

[13]  Davoud Mougouei,et al.  Factoring requirement dependencies in software requirement selection using graphs and integer programming , 2016, 2016 31st IEEE/ACM International Conference on Automated Software Engineering (ASE).

[14]  Moshe Sniedovich,et al.  Preference Order Stochastic Knapsack Problems: Methodological Issues , 1980 .

[15]  R. Burkard.  Quadratic Assignment Problems , 1984 .

[16]  Davoud Mougouei,et al.  Measuring Security of Web Services in Requirement Engineering Phase , 2012 .

[17]  John Scott.  What is social network analysis , 2010 .

[18]  John N. Mordeson,et al.  Applications of Fuzzy Graphs , 2000 .

[19]  David M. W. Powers,et al.  Dependency-Aware Software Release Planning , 2017, 2017 IEEE/ACM 39th International Conference on Software Engineering Companion (ICSE-C).