The static stochastic knapsack problem with normally distributed item sizes

• Published in: Mathematical programming Vol. 134; no. 2; pp. 459 - 489
• Main Authors: Merzifonluoğlu, Yasemin, Geunes, Joseph, Romeijn, H. Edwin
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer-Verlag 01.09.2012; Springer; Springer Nature B.V
• Subjects: Algorithms; Analysis; Applied sciences; Approximation; Calculus of Variations and Optimal Control; Optimization; Combinatorics; Exact sciences and technology; Exact solutions; Expected values; Flows in networks. Combinatorial problems; Full Length Paper; Heuristic; Knapsack problem; Mathematical analysis; Mathematical and Computational Physics; Mathematical Methods in Physics; Mathematical models; Mathematical programming; Mathematics; Mathematics and Statistics; Mathematics of Computing; Numerical Analysis; Operational research and scientific management; Operational research. Management science; Optimization; Production capacity; Random variables; Salvage; Salvage value; Stochasticity; Studies; Theoretical; 90C11; 90C26; 90C30; Overflow(computer arithmetics); Non convex programming; Branch and bound method; Gaussian distribution; Variable cost; Mixed integer programming; Non linear programming; Combinatorial optimization; Modeling; Knapsack problem; Exact solution; Static problems; Efficiency; Implicit enumeration method; Heuristic method; Profit; Production capacity
• DOI: 10.1007/s10107-011-0443-5
• ISSN: 0025-5610

This paper develops exact and heuristic algorithms for a stochastic knapsack problem where items with random sizes may be assigned to a knapsack. An item’s value is given by the realization of the product of a random unit revenue and the random item size. When the realization of the sum of selected item sizes exceeds the knapsack capacity, a penalty cost is incurred for each unit of overflow, while our model allows for a salvage value for each unit of capacity that remains unused. We seek to maximize the expected net profit resulting from the assignment of items to the knapsack. Although the capacity is fixed in our core model, we show that problems with random capacity, as well as problems in which capacity is a decision variable subject to unit costs, fall within this class of problems as well. We focus on the case where item sizes are independent and normally distributed random variables, and provide an exact solution method for a continuous relaxation of the problem. We show that an optimal solution to this relaxation exists containing no more than two fractionally selected items, and develop a customized branch-and-bound algorithm for obtaining an optimal binary solution. In addition, we present an efficient heuristic solution method based on our algorithm for solving the relaxation and empirically show that it provides high-quality solutions.