Heuristic and Exact Algorithms for the Precedence-Constrained Knapsack Problem

The knapsack problem (KP) is generalized taking into account a precedence relation between items. Such a relation can be represented by means of a directed acyclic graph, where nodes correspond to items in a one-to-one way. As in ordinary KPs, each item is associated with profit and weight, the knapsack has a fixed capacity, and the problem is to determine the set of items to be included in the knapsack. However, each item can be adopted only when all of its predecessors have been included in the knapsack. The knapsack problem with such an additional set of constraints is referred to as the precedence-constrained knapsack problem (PCKP). We present some dynamic programming algorithms that can solve small PCKPs to optimality, as well as a preprocessing method to reduce the size of the problem. Combining these, we are able to solve PCKPs with up to 2000 items in less than a few minutes of CPU time.     

Pricing and revenue maximization over a multicommodity transportation network: the nonlinear demand case

This paper is concerned with the design of efficient exact and heuristic algorithms for addressing a bilevel network pricing problem where demand is a nonlinear function of travel cost. The exact method is based on the piecewise linear approximation of the demand function, yielding mixed integer programming formulations, while heuristic procedures are developed within a bilevel trust region framework.     

Extending Dantzig's bound to the bounded multiple-class binary Knapsack problem

 The bounded multiple-class binary knapsack problem is a variant of the knapsack problem where the items are partitioned into classes and the item weights in each class are a multiple of a class weight. Thus, each item has an associated multiplicity. The constraints consists of an upper bound on the total item weight that can be selected and upper bounds on the total multiplicity of items that can be selected in each class. The objective is to maximize the sum of the profits associated with the selected items. This problem arises as a sub-problem in a column generation approach to the cutting stock problem. A special case of this model, where item profits are restricted to be multiples of a class profit, corresponds to the problem obtained by transforming an integer knapsack problem into a 0-1 form. However, the transformation proposed here does not involve a duplication of solutions as the standard transformation typically does. The paper shows that the LP-relaxation of this model can be solved by a greedy algorithm in linear time, a result that extends those of Dantzig (1957) and Balas and Zemel (1980) for the 0-1 knapsack problem. Hence, one can derive exact algorithms for the multi-class binary knapsack problem by adapting existing algorithms for the 0-1 knapsack problem. Computational results are reported that compare solving a bounded integer knapsack problem by transforming it into a standard binary knapsack problem versus using the multiple-class model as a 0-1 form. Received: May 1998 / Accepted: February 2002-09-04 Published online: December 9, 2002 Key Words. Knapsack problem – integer programming – linear programming relaxation     

Solving knapsack sharing problems with general tradeoff functions

A knapsack sharing problem is a maximin or minimax mathematical programming problem with one or more knapsack constraints (an inequality constraint with all non-negative coefficients). All knapsack sharing algorithms to date have assumed that the objective function is composed of single variable functions called tradeoff functions which are strictly increasing continuous functions. This paper develops optimality conditions and algorithms for knapsack sharing problems with any type of continuous tradeoff function including multiple-valued and staircase functions which can be increasing, decreasing, unimodal, bimodal, or even multi-modal. To do this, optimality conditions are developed for a special type of multiple-valued, nondecreasing tradeoff function called an ascending function. The optimal solution to any knapsack sharing problem can then be found by solving an equivalent problem where all the tradeoff functions have been transformed to ascending functions. Polynomial algorithms are developed for piecewise linear tradeoff functions with a fixed number of breakpoints. The techniques needed to construct efficient algorithms for any type of tradeoff function are discussed.     

Knapsack-based Algorithms for Designing Cellular Manufacturing Systems

In this paper, we use a 0–1 linear programming model for the machine assignment problem in cellular manufacturing. The formulated machine assignment problem considers many manufacturing factors such as machine utilization cost, cost of intercellular moves, part demand, and operating time capacity. The problem is shown to be NP-complete. Two designing algorithms are proposed to exploit the problem structure of the knapsack problem. Computational experience of the two algorithms is also reported.     

Simple solution methods for separable mixed linear and quadratic knapsack problem

Quadratic knapsack problem has a central role in integer and nonlinear optimization, which has been intensively studied due to its immediate applications in many fields and theoretical reasons. Although quadratic knapsack problem can be solved using traditional nonlinear optimization methods, specialized algorithms are much faster and more reliable than the nonlinear programming solvers. In this paper, we study a mixed linear and quadratic knapsack with a convex separable objective function subject to a single linear constraint and box constraints. We investigate the structural properties of the studied problem, and develop a simple method for solving the continuous version of the problem based on bi-section search, and then we present heuristics for solving the integer version of the problem. Numerical experiments are conducted to show the effectiveness of the proposed solution methods by comparing our methods with some state of the art linear and quadratic convex solvers.     

Implementing an efficient fptas for the 0–1 multi-objective knapsack problem

In the present work, we are interested in the practical behavior of a new fully polynomial time approximation schemes (fptas) to solve the approximation version of the 0–1 multi-objective knapsack problem. The proposed methodology makes use of very general techniques (such as dominance relations in dynamic programming) and thus may be applicable in the implementation of fptas for other problems as well.     

The zero-one knapsack problem with equality constraint

The zero-one knapsack problem is a linear zero-one programming problem with a single inequality constraint. This problem has been extensively studied and many applications and efficient algorithms have been published. In this paper we consider a similar problem, one with an equality instead of the inequality constraint. By replacing the equality by two inequalities one of which is placed in the economic function, a Lagrangean relaxation of the problem is obtained. The relation between the relaxed problem and the original problem is examined and it is shown how the optimal value of the relaxed problem varies with increasing values of the Lagrangean multiplier. Using these results an algorithm for solving the problem is proposed.The paper concludes with a discussion of computational experience.     

Very Large-Scale Neighborhood Search for the K-Constraint Multiple Knapsack Problem

The K-Constraint Multiple Knapsack Problem (K-MKP) is a generalization of the multiple knapsack problem, which is one of the representative combinatorial optimization problems known to be NP-hard. In K-MKP, each item has K types of weights and each knapsack has K types of capacity. In this paper, we propose several very large-scale neighborhood search (VLSN) algorithms to solve K-MKP. One of the VLSN algorithms incorporates a novel approach that consists of randomly perturbing the current solution in order to efficiently produce a set of simultaneous non-profitable moves. These moves would allow several items to be transferred from their current knapsacks and assigned to new knapsacks, which makes room for new items to be inserted through multi-exchange movements and allows for improved solutions. Computational results presented show that the method is effective, and provides better solutions compared to exact algorithms run for the same amount of time. This paper was written during Dr. Cunha's sabbatical at the Industrial and Systems Engineering Department at the University of Florida, Gainesville as a visiting faculty     

A heuristic algorithm for a chance constrained stochastic program

A chance constrained stochastic program is considered that arises from an application to college enrollments and in which the objective function is the expectation of a linear function of the random variables. When these random variables are independent and normally distributed with mean and variance that are linear in the decision variables, the deterministic equivalent of the problem is a nonconvex nonlinear knapsack problem. The optimal solution to this problem is characterized and a greedy-type heuristic algorithm that exploits this structure is employed. Computational results show that the algorithm performs well, especially when the normal random variables are approximations of binomial random variables.     

Dynamic programming based algorithms for the discounted {0-1} knapsack problem

The discounted {0-1} knapsack problem (DKP) is an extension of the classical {0-1} knapsack problem (KP) that consists of selecting a set of item groups where each group includes three items and at most one of the three items can be selected. The DKP is more challenging than the KP because four choices of items in an item group diversify the selection of the items. Consequently, it is not possible to solve the DKP based on a classical definition of a core consisting of a small number of relevant variables. This paper partitions the DKP into several easier sub-problems to achieve problem reductions by imitating the core concept of the KP to derive an alternative core for the DKP. Numerical experiments with DP-based algorithms are conducted to evaluate the effectiveness of the problem partition by solving the partitioned problem and the original problem based on different types of DKP instances.     

A single-resource allocation problem with Poisson resource requirements

We consider a stochastic resource allocation problem that generalizes the knapsack problem to account for random item weights that follow a Poisson distribution. When the sum of realized weights exceeds capacity, a penalty cost is incurred. We wish to select the items that maximize expected profit. We provide an effective solution method and illustrate the advantages of this approach via computational experiments.     

Knapsack with variable weights satisfying linear constraints

We introduce a variant of the knapsack problem, in which the weights of items are also variables but must satisfy a system of linear constraints, and the capacity of knapsack is given and known. We discuss two models: (1) the value of each item is given; (2) the value-to-weight ratio of each item is given. The goal is to determine the weight of each item, and to find a subset of items such that the total weight is no more than the capacity and the total value is maximized. We provide several practical application scenarios that motivate our study, and then investigate the computational complexity and corresponding algorithms. In particular, we show that if the number of constraints is a fixed constant, then both problems can be solved in polynomial time. If the number of constraints is an input, then we show that the first problem is NP-Hard and cannot be approximated within any constant factor unless \(\mathrm{P}= \mathrm{NP}\), while the second problem is NP-Hard but admits a polynomial time approximation scheme. We further propose approximation algorithms for both problems, and extend the results to multiple knapsack cases with a fixed number of knapsacks and identical capacities.     

Particle swarm optimization with time-varying acceleration coefficients for the multidimensional knapsack problem

The multidimensional knapsack problem (MKP) is a difficult combinatorial optimization problem, which has been proven as NP-hard problems. Various population-based search algorithms are applied to solve these problems. The particle swarm optimization (PSO) technique is adapted in our study, which proposes two novel PSO algorithms, namely, the binary PSO with time-varying acceleration coefficients (BPSOTVAC) and the chaotic binary PSO with time-varying acceleration coefficients (CBPSOTVAC). The two proposed methods were tested using 116 benchmark problems from the OR-Library to validate and demonstrate the efficiency of these algorithms in solving multidimensional knapsack problems. The results were then compared with those in the other two existing PSO algorithms. The simulation and evaluation results showed that the proposed algorithms, BPSOTVAC and CBPSOTVAC, are superior over the other methods according to its success rate, mean absolute deviation, mean absolute percentage error, least error, and standard deviation.     

Algorithms for the bounded set-up knapsack problem

The Bounded Set-up Knapsack Problem (BSKP) is a generalization of the Bounded Knapsack Problem (BKP), where each item type has a set-up weight and a set-up value that are included in the knapsack and the objective function value, respectively, if any copies of that item type are in the knapsack. This paper provides three dynamic programming algorithms that solve BSKP in pseudo-polynomial time and a fully polynomial-time approximation scheme (FPTAS). A key implication from these results is that the dynamic programming algorithms and the FPTAS can also be applied to BKP. One of the dynamic programming algorithms presented solves BKP with the same time and space bounds of the best known dynamic programming algorithm for BKP. Moreover, the FPTAS improves the worst-case time bound for obtaining approximate solutions to BKP as compared to using FPTASs designed for BKP or the 0-1 Knapsack Problem.     

Test sets of integer programs

This article is a survey about recent developments in the area of test sets of families of linear integer programs. Test sets are finite subsets of the integer lattice that allow to improve any given feasible non-optimal point of an integer program by one element in the set. There are various possible ways of defining test sets depending on the view that one takes: theGraver test set is naturally derived from a study of the integral vectors in cones; theScarf test set (neighbors of the origin) is strongly connected to the study of lattice point free convex bodies; the so-calledreduced Gröbner basis of an integer program is obtained from a study of generators of polynomial ideals. This explains why the study of test sets connects various branches of mathematics. We introduce in this paper these three kinds of test sets and discuss relations between them. We also illustrate on various examples such as the minimum cost flow problem, the knapsack problem and the matroid optimization problem how these test sets may be interpreted combinatorially. From the viewpoint of integer programming a major interest in test sets is their relation to the augmentation problem. This is discussed here in detail. In particular, we derive a complexity result of the augmentation problem, we discuss an algorithm for solving the augmentation problem by computing the Graver test set and show that, in the special case of an integer knapsack problem with 3 coefficients, the augmentation problem can be solved in polynomial time.Supported by a Gerhard-Hess-Forschungsförderpreis of the German Science Foundation (DFG).     

A reduction dynamic programming algorithm for the bi-objective integer knapsack problem

This paper presents a backward state reduction dynamic programming algorithm for generating the exact Pareto frontier for the bi-objective integer knapsack problem. The algorithm is developed addressing a reduced problem built after applying variable fixing techniques based on the core concept. First, an approximate core is obtained by eliminating dominated items. Second, the items included in the approximate core are subject to the reduction of the upper bounds by applying a set of weighted-sum functions associated with the efficient extreme solutions of the linear relaxation of the multi-objective integer knapsack problem. Third, the items are classified according to the values of their upper bounds; items with zero upper bounds can be eliminated. Finally, the remaining items are used to form a mixed network with different upper bounds. The numerical results obtained from different types of bi-objective instances show the effectiveness of the mixed network and associated dynamic programming algorithm.     

A class of nonlinear nonseparable continuous knapsack and multiple-choice knapsack problems

This paper considers a general class of continuous, nonlinear, and nonseparable knapsack problems, special cases of which arise in numerous operations and financial contexts. We develop important properties of optimal solutions for this problem class, based on the properties of a closely related class of linear programs. Using these properties, we provide a solution method that runs in polynomial time in the number of decision variables, while also depending on the time required to solve a particular one-dimensional optimization problem. Thus, for the many applications in which this one-dimensional function is reasonably well behaved (e.g., unimodal), the resulting algorithm runs in polynomial time. We next develop a related solution approach to a class of continuous, nonlinear, and nonseparable multiple-choice knapsack problems. This algorithm runs in polynomial time in both the number of variables and the number of variants per item, while again dependent on the complexity of the same one-dimensional optimization problem as for the knapsack problem. Computational testing demonstrates the power of the proposed algorithms over a commercial global optimization software package.     

Fractional knapsack problems

The fractional knapsack problem to obtain an integer solution that maximizes a linear fractional objective function under the constraint of one linear inequality is considered. A modification of the Dinkelbach's algorithm [3] is proposed to exploit the fact that good feasible solutions are easily obtained for both the fractional knapsack problem and the ordinary knapsack problem. An upper bound of the number of iterations is derived. In particular it is clarified how optimal solutions depend on the right hand side of the constraint; a fractional knapsack problem reduces to an ordinary knapsack problem if the right hand side exceeds a certain bound.     

The generalized assignment problem: Valid inequalities and facets

Three classes of valid inequalities based upon multiple knapsack constraints are derived for the generalized assignment problem. General properties of the facet defining inequalities are discussed and, for a special case, the convex hull is completely characterized. In addition, we prove that a basic fractional solution to the linear programming relaxation can be eliminated by a facet defining inequality associated with an individual knapsack constraint.Partial financial support under NSF grant #CCR-8812736.Partial financial support under NSF grant #DMS-8606188.