Knapsack problem with probability constraints

This paper is dedicated to a study of different extensions of the classical knapsack problem to the case when different elements of the problem formulation are subject to a degree of uncertainty described by random variables. This brings the knapsack problem into the realm of stochastic programming. Two different model formulations are proposed, based on the introduction of probability constraints. The first one is a static quadratic knapsack with a probability constraint on the capacity of the knapsack. The second one is a two-stage quadratic knapsack model, with recourse, where we introduce a probability constraint on the capacity of the knapsack in the second stage. As far as we know, this is the first time such a constraint has been used in a two-stage model. The solution techniques are based on the semidefinite relaxations. This allows for solving large instances, for which exact methods cannot be used. Numerical experiments on a set of randomly generated instances are discussed below. Introduction The knapsack problem (KP) is a well-known and well-studied problem in combinatorial optimization. Knapsack problems are often used to model industrial situations or financial decisions. They may also appear as sub-problems of larger or more complex problems. The most famous form of KP is the single constraint binary version: we are given N items, with a profit pi for the item i, and a weight wi for the item i, with i = 1, ..., N , and a knapsack capacity c. The problem is to select a subset of items so that the weight of the subset does not exceed c, and returns a maximal total profit. In this form, the problem is known to be NP -Hard [GJ79], and has been intensively studied in the past decades, and we now know several exact and approximate algorithms for this problem. In particular, it admits a FPTAS [IK75]. For the quadratic knapsack problem, a survey done by David Pisinger [Pis07] gives detailed information on the problem and a number of results on the performance of various relaxations and algorithms used to solve or approximate the problem. In the case of modeling financial decisions, transportation, or production plans, however, this formulation shows its limitations, since it does not take into account uncertainty on the problem parameters, such as the prices pi or the weights wj . Similarly, such decisions are not static, and this model cannot take into account new information available on the prices or the weights. There have been several studies done on the stochastic knapsack problem in the past years: work has been done to find heuristics [CB98], approximation algorithms [KP98], [DGV04], [SS06]. Department of Industrial Economy and Technology Management, Norwegian University of Science and Technology, Trondheim, Norway. Laboratoire de Recherche en Informatique, Université de Paris Sud, 91405, Orsay Cedex, France. 1 Stochastic knapsack problems can reach a number of binary variables and constraints of such magnitude that commercial packages cannot find a solution in a reasonable time or memory space, requiring the use of linear relaxations to find an upper bound on the problem. While linear relaxations were successful for many combinatorial optimization problems, it turns out that knapsack problems, especially their quadratic formulation, can not be approximated tightly by linearization based methods. Stronger relaxation methods, namely semidefinite relaxations (called SDP thereafter), have turned out to be particularly interesting for such combinatorial optimization problems [GW95], [HR98]. See also [Pis07] for a survey on the quadratic knapsack problem. More precisely, semidefinite programming is a recent development of convex optimization, which deals with optimization problems over symmetric positive semidefinite matrices with linear cost function and linear constraints. Groetschel and al. showed that semidefinite optimization problems can be solved in polynomial time [GLS88]. In this paper we present two variants of stochastic optimization problems: the first one is a static quadratic knapsack problem with probability constraint. The probability constraint is used to model the risk we are willing to take when making our decision. We only know some information about the weights of the items, but we have to take a decision with this limited knowledge at the risk of breaking the capacity constraint. The second model is the stochastic quadratic knapsack problem with recourse. In this model, we have a two-stage formulation which models a situation where we make a decision with limited information, but where a second decision (the recourse) can be made afterwards, after receiving information about the weight, or prices. This allows to modify initial decisions. In this model, since at the second step, some information is still unknown, we face a risk of breaking the capacity constraint when making the decision. We can view the static stochastic knapsack as a resource allocation problem. A planner has to choose among various products, various foods for humanitarian relief for instance, and has to choose among them a subset to fill crates for maximal utility, with the constraint that the crates can only carry a finite weight of food. However, he does not have all the information to select the best products since the food has not yet been collected. With this limited information, a decision can still be made where we accept a certain low risk, or probability, of selecting more products than can fit in the crate. The case with recourse would then be making a decision with part of the supplies known, and making changes in the food that will be packed once more information about the rest of the food has been collected. However, there is a cost to move the food in or out of the crate at the last moment. Again, since some of the information is not known, we also face a small risk of planning to try to fit too many products into the crate. This paper is organized as follows: first, we present the static knapsack problem with probability constraint wherein we give a presentation of the problem. Then we reformulate this problem under the form of an SDP problem and detail two different relaxations. We then present the stochastic knapsack problem with recourse. Finally, numerical experiments are given in a third section. 1 Static quadratic knapsack problem with probability constraint In this section, we present the static knapsack problem with a probability constraint. We formulate the uncertainty over the weights wi with random variables, and we consider a probability (1−α) of satisfying the constraints, where α is the risk measurement. We then present the SDP relaxation, and the two relaxations we used to solve our problem.