A note on the solution of group knapsack problems

Using the Nicholson principle the algorithm of Shapiro for solving group knapsack problems is improved. An approximation method is derived and numerical results are presented. The solution of the approximation method will be characterized.     

A new lower bound for the linear knapsack problem with general integer variables

It is well known that the linear knapsack problem with general integer variables (LKP) is NP-hard. In this paper we first introduce a special case of this problem and develop an O(n) algorithm to solve it. We then show how this algorithm can be used efficiently to obtain a lower bound for a general instance of LKP and prove that it is at least as good as the linear programming lower bound. We also present the results of a computational study that show that for certain classes of problems the proposed bound on average is tighter than other bounds proposed in the literature.     

An exact algorithm for the fixed-charge multiple knapsack problem

We formulate the fixed-charge multiple knapsack problem (FCMKP) as an extension of the multiple knapsack problem (MKP). The Lagrangian relaxation problem is easily solved, and together with a greedy heuristic we obtain a pair of upper and lower bounds quickly. We make use of these bounds in the pegging test to reduce the problem size. We also present a branch-and-bound (B&B) algorithm to solve FCMKP to optimality. This algorithm exploits the Lagrangian upper bound as well as the pegging result for pruning, and at each terminal subproblem solve MKP exactly by invoking MULKNAP code developed by Pisinger [Pisinger, D., 1999. An exact algorithm for large multiple knapsack problems. European Journal of Operational Research 114, 528–541]. As a result, we are able to solve almost all test problems with up to 32,000 items and 50 knapsacks within a few seconds on an ordinary computing environment, although the algorithm remains some weakness for small instances with relatively many knapsacks.     

Simultaneously lifting sets of binary variables into cover inequalities for knapsack polytopes

Cover inequalities are commonly used cutting planes for the 0–1 knapsack problem. This paper describes a linear-time algorithm (assuming the knapsack is sorted) to simultaneously lift a set of variables into a cover inequality. Conditions for this process to result in valid and facet-defining inequalities are presented. In many instances, the resulting simultaneously lifted cover inequality cannot be obtained by sequentially lifting over any cover inequality. Some computational results demonstrate that simultaneously lifted cover inequalities are plentiful, easy to find and can be computationally beneficial.     

Nonconvex piecewise linear knapsack problems

This paper considers the minimization version of a class of nonconvex knapsack problems with piecewise linear cost structure. The items to be included in the knapsack have a divisible quantity and a cost function. An item can be included partially in the given quantity range and the cost is a nonconvex piecewise linear function of quantity. Given a demand, the optimization problem is to choose an optimal quantity for each item such that the demand is satisfied and the total cost is minimized. This problem and its close variants are encountered in manufacturing planning, supply chain design, volume discount procurement auctions, and many other contemporary applications. Two separate mixed integer linear programming formulations of this problem are proposed and are compared with existing formulations. Motivated by different scenarios in which the problem is useful, the following algorithms are developed: (1) a fast polynomial time, near-optimal heuristic using convex envelopes; (2) exact pseudo-polynomial time dynamic programming algorithms; (3) a 2-approximation algorithm; and (4) a fully polynomial time approximation scheme. A comprehensive test suite is developed to generate representative problem instances with different characteristics. Extensive computational experiments show that the proposed formulations and algorithms are faster than the existing techniques.     

A knapsack intersection hierarchy

We introduce a knapsack intersection hierarchy for strengthening linear programming relaxations of packing integer programs. In level t of the hierarchy, all valid cuts are added for the integer hull of the intersection of all t-row relaxations. This model captures the maximum possible strength of t-row cuts, an approach often used by solvers for small t. We investigate the integrality gap of the strengthened formulations on the all-or-nothing flow problem in trees (also called unsplittable flow on trees).     

Revisiting surrogate relaxation for the multidimensional knapsack problem

The multidimensional knapsack problem (MKP) is a classic problem in combinatorial optimisation. Several authors have proposed to use surrogate relaxation to compute upper bounds for the MKP. In their papers, the surrogate dual is solved heuristically. We show that, using a modern dual simplex solver as a subroutine, one can solve the surrogate dual exactly in reasonable computing times. On the other hand, the resulting upper bound tends to be strong only for relatively small MKP instances.     

An O(n) algorithm for quadratic knapsack problems

An algorithm is presented which solves bounded quadratic optimization problems with n variables and one linear constraint in at most O(n) steps. The algorithm is based on a parametric approach combined with well-known ideas for constructing efficient algorithms. It improves an O(n log n) algorithm which has been developed for a more restricted case of the problem.     

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.     

A dynamic programming based reduction procedure for the multidimensional 0–1 knapsack problem

This paper presents a preprocessing procedure for the 0–1 multidimensional knapsack problem. First, a non-increasing sequence of upper bounds is generated by solving LP-relaxations. Then, a non-decreasing sequence of lower bounds is built using dynamic programming. The comparison of the two sequences allows either to prove that the best feasible solution obtained is optimal, or to fix a subset of variables to their optimal values. In addition, a heuristic solution is obtained. Computational experiments with a set of large-scale instances show the efficiency of our reduction scheme. Particularly, it is shown that our approach allows to reduce the CPU time of a leading commercial software.     

Greedy algorithm for the general multidimensional knapsack problem

In this paper, we propose a new greedy-like heuristic method, which is primarily intended for the general MDKP, but proves itself effective also for the 0-1 MDKP. Our heuristic differs from the existing greedy-like heuristics in two aspects. First, existing heuristics rely on each item’s aggregate consumption of resources to make item selection decisions, whereas our heuristic uses the effective capacity, defined as the maximum number of copies of an item that can be accepted if the entire knapsack were to be used for that item alone, as the criterion to make item selection decisions. Second, other methods increment the value of each decision variable only by one unit, whereas our heuristic adds decision variables to the solution in batches and consequently improves computational efficiency significantly for large-scale problems. We demonstrate that the new heuristic significantly improves computational efficiency of the existing methods and generates robust and near-optimal solutions. The new heuristic proves especially efficient for high dimensional knapsack problems with small-to-moderate numbers of decision variables, usually considered as “hard” MDKP and no computationally efficient heuristic is available to treat such problems. Supported in part by the NSF grant DMI 9812994.     

Stochastic on-line knapsack problems

Different classes of on-line algorithms are developed and analyzed for the solution of {0, 1} and relaxed stochastic knapsack problems, in which both profit and size coefficients are random variables. In particular, a linear time on-line algorithm is proposed for which the expected difference between the optimum and the approximate solution value isO(log^3/2 n). An(1) lower bound on the expected difference between the optimum and the solution found by any on-line algorithm is also shown to hold.Corresponding author.Partially supported by the Basic Research Action of the European Communities under Contract 3075 (Alcom).Partially supported by research project Models and Algorithms for Optimization of the Italian Ministry of University and Scientific and Technological Research (MURST 40%).     

Efficient algorithms for solving multiconstraint zero-one knapsack problems to optimality

The multiconstraint 0–1 knapsack problem is encountered when one has to decide how to use a knapsack with multiple resource constraints. Even though the single constraint version of this problem has received a lot of attention, the multiconstraint knapsack problem has been seldom addressed. This paper deals with developing an effective solution procedure for the multiconstraint knapsack problem. Various relaxation of the problem are suggested and theoretical relations between these relaxations are pointed out. Detailed computational experiments are carried out to compare bounds produced by these relaxations. New algorithms for obtaining surrogate bounds are developed and tested. Rules for reducing problem size are suggested and shown to be effective through computational tests. Different separation, branching and bounding rules are compared using an experimental branch and bound code. An efficient branch and bound procedure is developed, tested and compared with two previously developed optimal algorithms. Solution times with the new procedure are found to be considerably lower. This procedure can also be used as a heuristic for large problems by early termination of the search tree. This scheme was tested and found to be very effective.     

A branch and search algorithm for a class of nonlinear knapsack problems

This paper discusses a class of nonlinear knapsack problems where the objective function is quadratic. The method is a branch and search procedure which includes an efficient algorithm to find the continuous (relaxed) solution and a reduction rule which computes tight lower and upper bounds on the integer variables.     

The 0-1 knapsack problem with fuzzy data

The 0-1 knapsack problem with imprecise profits and imprecise weights of items is considered. The imprecise parameters are modeled as fuzzy intervals. A method of choosing a solution under the uncertainty is proposed and two methods for solving the constructed models are provided.     

A dynamic programming algorithm for the bilevel knapsack problem

We propose an efficient dynamic programming algorithm for solving a bilevel program where the leader controls the capacity of a knapsack, and the follower solves the resulting knapsack problem. We propose new recursive rules and show how to solve the problem as a sequence of two standard knapsack problems.     

A unified method for a class of convex separable nonlinear knapsack problems

In this paper, a unified algorithm is proposed for solving a class of convex separable nonlinear knapsack problems, which are characterized by positive marginal cost (PMC) and increasing marginal loss–cost ratio (IMLCR). By taking advantage of these two characteristics, the proposed algorithm is applicable to the problem with equality or inequality constraints. In contrast to the methods based on Karush–Kuhn–Tucker (KKT) conditions, our approach has linear computation complexity. Numerical results are reported to demonstrate the efficacy of the proposed algorithm for different problems.     

Solving the knapsack problem with imprecise weight coefficients using genetic algorithms

This paper investigates solving the knapsack problem with imprecise weight coefficients using genetic algorithms. This work is based on the assumption that each weight coefficient is imprecise due to decimal truncation or coefficient rough estimation by the decision-maker. To deal with this kind of imprecise data, fuzzy sets provide a powerful tool to model and solve this problem. We investigate the possibility of using genetic algorithms in solving the fuzzy knapsack problem without defining membership functions for each imprecise weight coefficient. The proposed approach simulates a fuzzy number by distributing it into some partition points. We use genetic algorithms to evolve the values in each partition point so that the final values represent the membership grade of a fuzzy number. The empirical results show that the proposed approach can obtain very good solutions within the given bound of each imprecise weight coefficient than the fuzzy knapsack approach. The fuzzy genetic algorithm concept approach is different, but gives better results than the traditional fuzzy approach.     

Smart greedy procedure for solving a nOnlinear knapsack class of reliability optimization problems

A new heuristic procedure, which is called Smart Greedy, is proposed for solving a kind of general reliability optimization problems (non-DGR type knapsack problems). Smart Greedy uses Recursive Greedy with multiple greedy functions designated by balance coefficients, generates several solutions and then determines the best solution among them as the smart greedy solution. Recursive Greedy first checks the feasibility of sets of items for a given problem and removes infeasible items from the item sets. Second, the procedure checks the gain ratio of increments of objective function to constraint function and reduces the problem to DGR type problem by invoking LP dominance. Third, the procedure continues to allocate the increments for current items until the constraint is violated. With the current solution, the procedure then repeats the greedy procedure for current items that are added to the items removed by the LP dominance in the previous step.Computational results show that the Smart Greedy is more effective than the previously reported methods.     

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.