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.     

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.     

New convergent heuristics for 0–1 mixed integer programming

Several hybrid methods have recently been proposed for solving 0–1 mixed integer programming problems. Some of these methods are based on the complete exploration of small neighborhoods. In this paper, we present several convergent algorithms that solve a series of small sub-problems generated by exploiting information obtained from a series of relaxations. These algorithms generate a sequence of upper bounds and a sequence of lower bounds around the optimal value. First, the principle of a linear programming-based algorithm is summarized, and several enhancements of this algorithm are presented. Next, new hybrid heuristics that use linear programming and/or mixed integer programming relaxations are proposed. The mixed integer programming (MIP) relaxation diversifies the search process and introduces new constraints in the problem. This MIP relaxation also helps to reduce the gap between the final upper bound and lower bound. Our algorithms improved 14 best-known solutions from a set of 108 available and correlated instances of the 0–1 multidimensional Knapsack problem. Other encouraging results obtained for 0–1 MIP problems are also presented.     

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     

An exact approach for solving integer problems under probabilistic constraints with random technology matrix

This paper addresses integer programming problems under probabilistic constraints involving discrete distributions. Such problems can be reformulated as large scale integer problems with knapsack constraints. For their solution we propose a specialized Branch and Bound approach where the feasible solutions of the knapsack constraint are used as partitioning rules of the feasible domain. The numerical experience carried out on a set covering problem with random covering matrix shows the validity of the solution approach and the efficiency of the implemented algorithm.     

Estimation of the number of iterations in integer programming algorithms using the regular partitions method

We review the results of studying integer linear programming algorithms which exploit properties of problem relaxation sets. The main attention is paid to the estimation of the number of iterations of these algorithms by means of the regular partitions method and other approaches. We present such estimates for some cutting plane, branch and bound (Land and Doig scheme), and L-class enumeration algorithms and consider questions of their stability. We establish the upper bounds for the average number of iterations of the mentioned algorithms as applied to the knapsack problem and the set packing one.     

A column generation based decomposition algorithm for a parallel machine just-in-time scheduling problem

We propose a column generation based exact decomposition algorithm for the problem of scheduling n jobs with an unrestrictively large common due date on m identical parallel machines to minimize total weighted earliness and tardiness. We first formulate the problem as an integer program, then reformulate it, using Dantzig–Wolfe decomposition, as a set partitioning problem with side constraints. Based on this set partitioning formulation, a branch and bound exact solution algorithm is developed for the problem. In the branch and bound tree, each node is the linear relaxation problem of a set partitioning problem with side constraints. This linear relaxation problem is solved by column generation approach where columns represent partial schedules on single machines and are generated by solving two single machine subproblems. Our computational results show that this decomposition algorithm is capable of solving problems with up to 60 jobs in reasonable cpu time.     

Exact and heuristic solution approaches for the mixed integer setup knapsack problem

We consider a class of knapsack problems that include setup costs for families of items. An individual item can be loaded into the knapsack only if a setup cost is incurred for the family to which it belongs. A mixed integer programming formulation for the problem is provided along with exact and heuristic solution methods. The exact algorithm uses cross decomposition. The proposed heuristic gives fast and tight bounds. In addition, a Benders decomposition algorithm is presented to solve the continuous relaxation of the problem. This method for solving the continuous relaxation can be used to improve the performance of a branch and bound algorithm for solving the integer problem. Computational performance of the algorithms are reported and compared to CPLEX.     

不可分离凸背包问题的拉格朗日分解和区域分割方法

本文对线性约束不可分离凸背包问题给出了一种精确算法．该算法是拉格朗日分解和区域分割结合起来的一种分枝定界算法．利用拉格朗日分解方法可以得到每个子问题的一个可行解，一个不可行解，一个下界和一个上界．区域分割可以把一个整数箱子分割成几个互不相交的整数子箱子的并集，每个整数子箱子对应一个子问题．通过区域分割可以逐步减小对偶间隙并最终经过有限步迭代找到原问题的最优解．数值结果表明该算法对不可分离凸背包问题是有效的．     

Resolution method for mixed integer bi-level linear problems based on decomposition technique

In this article, we propose a new algorithm for the resolution of mixed integer bi-level linear problem (MIBLP). The algorithm is based on the decomposition of the initial problem into the restricted master problem (RMP) and a series of problems named slave problems (SP). The proposed approach is based on Benders decomposition method where in each iteration a set of variables are fixed which are controlled by the upper level optimization problem. The RMP is a relaxation of the MIBLP and the SP represents a restriction of the MIBLP. The RMP interacts in each iteration with the current SP by the addition of cuts produced using Lagrangian information from the current SP. The lower and upper bound provided from the RMP and SP are updated in each iteration. The algorithm converges when the difference between the upper and lower bound is within a small difference ε. In the case of MIBLP Karush–Kuhn–Tucker (KKT) optimality conditions could not be used directly to the inner problem in order to transform the bi-level problem into a single level problem. The proposed decomposition technique, however, allows the use of KKT conditions and transforms the MIBLP into two single level problems. The algorithm, which is a new method for the resolution of MIBLP, is illustrated through a modified numerical example from the literature. Additional examples from the literature are presented to highlight the algorithm convergence properties.     

A partitioning algorithm for the network loading problem

This paper proposes a Benders-like partitioning algorithm to solve the network loading problem. The approach is an iterative method in which the integer programming solver is not used to produce the best integer point in the polyhedral relaxation of the set of feasible capacities. Rather, it selects an integer solution that is closest to the best known integer solution. Contrary to previous approaches, the method does not exploit the original mixed integer programming formulation of the problem. The effort of computing integer solutions is entirely left to a pure integer programming solver while valid inequalities are generated by solving standard nonlinear multicommodity flow problems. The method is compared to alternative approaches proposed in the literature and appears to be efficient for computing good upper bounds.     

A column generation heuristic for the two-dimensional two-staged guillotine cutting stock problem with multiple stock size

We consider a two-dimensional cutting stock problem where stock of different sizes is available, and a set of rectangular items has to be obtained through two-staged guillotine cuts. We propose a heuristic algorithm, based on column generation, which requires as its subproblem the solution of a two-dimensional knapsack problem with two-staged guillotines cuts. A further contribution of the paper consists in the definition of a mixed integer linear programming model for the solution of this knapsack problem, as well as a heuristic procedure based on dynamic programming. Computational experiments show the effectiveness of the proposed approach, which obtains very small optimality gaps and outperforms the heuristic algorithm proposed by Cintra et al. [3].     

A semidefinite programming method for integer convex quadratic minimization

We consider the NP-hard problem of minimizing a convex quadratic function over the integer lattice \({\mathbf{Z}}^n\). We present a simple semidefinite programming (SDP) relaxation for obtaining a nontrivial lower bound on the optimal value of the problem. By interpreting the solution to the SDP relaxation probabilistically, we obtain a randomized algorithm for finding good suboptimal solutions, and thus an upper bound on the optimal value. The effectiveness of the method is shown for numerical problem instances of various sizes.     

Zero-one integer programs with few constraints —Lower bounding theory

The zero-one integer programming problem and its special case, the multiconstraint knapsack problem frequently appear as subproblems in many combinatorial optimization problems. We present several methods for computing lower bounds on the optimal solution of the zero-one integer programming problem. They include Lagrangean, surrogate and composite relaxations. New heuristic procedures are suggested for determining good surrogate multipliers. Based on theoretical results and extensive computational testing, it is shown that for zero-one integer problems with few constraints surrogate relaxation is a viable alternative to the commonly used Lagrangean and linear programming relaxations. These results are used in a follow up paper to develop an efficient branch and bound algorithm for solving zero-one integer programming problems.     

A “reduce and solve” approach for the multiple-choice multidimensional knapsack problem

The multiple-choice multidimensional knapsack problem (MMKP) is a well-known NP-hard combinatorial optimization problem with a number of important applications. In this paper, we present a “reduce and solve” heuristic approach which combines problem reduction techniques with an Integer Linear Programming (ILP) solver (CPLEX). The key ingredient of the proposed approach is a set of group fixing and variable fixing rules. These fixing rules rely mainly on information from the linear relaxation of the given problem and aim to generate reduced critical subproblem to be solved by the ILP solver. Additional strategies are used to explore the space of the reduced problems. Extensive experimental studies over two sets of 37 MMKP benchmark instances in the literature show that our approach competes favorably with the most recent state-of-the-art algorithms. In particular, for the set of 27 conventional benchmarks, the proposed approach finds an improved best lower bound for 11 instances and as a by-product improves all the previous best upper bounds. For the 10 additional instances with irregular structures, the method improves 7 best known results.     

A branch and bound algorithm for the maximum diversity problem

This article begins with a review of previously proposed integer formulations for the maximum diversity problem (MDP). This problem consists of selecting a subset of elements from a larger set in such a way that the sum of the distances between the chosen elements is maximized. We propose a branch and bound algorithm and develop several upper bounds on the objective function values of partial solutions to the MDP. Empirical results with a collection of previously reported instances indicate that the proposed algorithm is able to solve all the medium-sized instances (with 50 elements) as well as some large-sized instances (with 100 elements). We compare our method with the best previous linear integer formulation solved with the well-known software Cplex. The comparison favors the proposed procedure.     

Linear programming for the 0–1 quadratic knapsack problem

In this paper we consider the quadratic knapsack problem which consists in maximizing a positive quadratic pseudo-Boolean function subject to a linear capacity constraint. We propose a new method for computing an upper bound. This method is based on the solution of a continuous linear program constructed by adding to a classical linearization of the problem some constraints rebundant in 0–1 variables but nonredundant in continuous variables. The obtained upper bound is better than the bounds given by other known methods. We also propose an algorithm for computing a good feasible solution. This algorithm is an elaboration of the heuristic methods proposed by Chaillou, Hansen and Mahieu and by Gallo, Hammer and Simeone. The relative error between this feasible solution and the optimum solution is generally less than 1%. We show how these upper and lower bounds can be efficiently used to determine the values of some variables at the optimum. Finally we propose a branch-and-bound algorithm for solving the quadratic knapsack problem and report extensive computational tests.     

Selection of Capacity Expansion Projects

The problem of determining a project selection schedule and a production-distribution-inventory schedule for each of a number of plants so as to meet the demands of multiregional markets at minimum discounted total cost during a discrete finite planning horizon is considered. We include the possibility of using inventory and/or imports to delay the expansion decision at each producing region in a transportation network. Through a problem reduction algorithm, the Lagrangean relaxation problem strengthened by the addition of a surrogate constraint becomes a 0–1 mixed integer knapsack problem. Its optimal solution, given a set of Lagrangean multipliers, can be obtained by solving at most two generally smaller 0–1 pure integer knapsack problems. The bound is usually very tight. At each iteration of the subgradient method, we generate a primal feasible solution from the Lagrangean solution. The computational results indicate that the procedure is effective in solving large problems to within acceptable error tolerances.     

Capacity and tool allocation problem in flexible manufacturing systems

This study addresses an allocation problem that arises in the semiconductor industry and flexible manufacturing systems where the tools should be loaded on computer numerical controlled (CNC) machines to process a number of operations. The time and tool magazine capacities of the CNC machines and the number of available tools of each type are limited. The objective is to maximize the total weight of operation assignments. We present a mixed integer programming formulation of the problem and show that the problem is NP-hard in the strong sense. We show that the linear programming relaxation upper bound dominates the best possible Lagrangean relaxation upper bound. We develop several lower bounding procedures and a lower bounding procedure using Lagrangean relaxation approach. Our computational results show that the upper and lower bounding procedures produce near-optimal solutions in reasonable times.     

A generalized knapsack problem with variable coefficients

The ordinary knapsack problem is to find the optimal combination of items to be packed in a knapsack under a single constraint on the total allowable resources, where all coefficients in the objective function and in the constraint are constant.In this paper, a generalized knapsack problem with coefficients depending on variable parameters is proposed and discussed. Developing an effective branch and bound algorithm for this problem, the concept of relaxation and the efficiency function introduced here will play important roles. Furthermore, a relation between the algorithm and the dynamic programming approach is discussed, and subsequently it will be shown that the ordinary 0–1 knapsack problem, the linear programming knapsack problem and the single constrained linear programming problem with upper-bounded variables are special cases of the interested problem. Finally, practical applications of the problem and its computational experiences will be shown.