Lagrangean relaxation for a lower bound to a set partitioning problem with side constraints: properties and algorithms

The problem (P) addressed here is a special set partitioning problem with two additional non-trivial constraints. A Lagrangean Relaxation (LR_u) is proposed to provide a lower bound to the optimal solution to this problem. This Lagrangean relaxation is accomplished by a subgradient optimization procedure which solves at each iteration a special 0–1 knapsack problem (KP-k). We give two procedures to solve (KP-k), namely an implicity enumeration algorithm and a column generation method. The approach is promising for it provides feasible integer solutions to the side constraints that will hopefully be optimal to most of the instances of the problem (P). Properties of the feasible solutions to (KP-k) are highlighted and it is shown that the linear programming relaxation to this problem has a worst case time bound of order O(n³).     

n-step mingling inequalities: new facets for the mixed-integer knapsack set

The n-step mixed integer rounding (MIR) inequalities of Kianfar and Fathi (Math Program 120(2):313–346, 2009) are valid inequalities for the mixed-integer knapsack set that are derived by using periodic n-step MIR functions and define facets for group problems. The mingling and 2-step mingling inequalities of Atamtürk and Günlük (Math Program 123(2):315–338, 2010) are also derived based on MIR and they incorporate upper bounds on the integer variables. It has been shown that these inequalities are facet-defining for the mixed integer knapsack set under certain conditions and generalize several well-known valid inequalities. In this paper, we introduce new classes of valid inequalities for the mixed-integer knapsack set with bounded integer variables, which we call n-step mingling inequalities (for positive integer n). These inequalities incorporate upper bounds on integer variables into n-step MIR and, therefore, unify the concepts of n-step MIR and mingling in a single class of inequalities. Furthermore, we show that for each n, the n-step mingling inequality defines a facet for the mixed integer knapsack set under certain conditions. For n?=?2, we extend the results of Atamtürk and Günlük on facet-defining properties of 2-step mingling inequalities. For n ≥ 3, we present new facets for the mixed integer knapsack set. As a special case we also derive conditions under which the n-step MIR inequalities define facets for the mixed integer knapsack set. In addition, we show that n-step mingling can be used to generate new valid inequalities and facets based on covers and packs defined for mixed integer knapsack sets.     

A Lagrangian relaxation approach to the edge-weighted clique problem

The b-clique polytope CPⁿ_b is the convex hull of the node and edge incidence vectors of all subcliques of size at most b of a complete graph on n nodes. Including the Boolean quadric polytope QPⁿ=CPⁿ_n as a special case and being closely related to the quadratic knapsack polytope, it has received considerable attention in the literature. In particular, the max-cut problem is equivalent with optimizing a linear function over CPⁿ_n. The problem of optimizing linear functions over CPⁿ_b has so far been approached via heuristic combinatorial algorithms and cutting-plane methods.We study the structure of CPⁿ_b in further detail and present a new computational approach to the linear optimization problem based on the idea of integrating cutting planes into a Lagrangian relaxation of an integer programming problem that Balas and Christofides had suggested for the traveling salesman problem. In particular, we show that the separation problem for tree inequalities becomes polynomial in our Lagrangian framework. Finally, computational results are presented.     

Reoptimizing the 0-1 knapsack problem

In this paper we study the problem where an optimal solution of a knapsack problem on n items is known and a very small number k of new items arrive. The objective is to find an optimal solution of the knapsack problem with n+k items, given an optimal solution on the n items (reoptimization of the knapsack problem). We show that this problem, even in the case k=1, is NP-hard and that, in order to have effective heuristics, it is necessary to consider not only the items included in the previously optimal solution and the new items, but also the discarded items. Then, we design a general algorithm that makes use, for the solution of a subproblem, of an α-approximation algorithm known for the knapsack problem. We prove that this algorithm has a worst-case performance bound of , which is always greater than α, and therefore that this algorithm always outperforms the corresponding α-approximation algorithm applied from scratch on the n+k items. We show that this bound is tight when the classical Ext-Greedy algorithm and the algorithm are used to solve the subproblem. We also show that there exist classes of instances on which the running time of the reoptimization algorithm is smaller than the running time of an equivalent PTAS and FPTAS.     

An n-tet graph approach for non-guillotine packings of n-dimensional boxes into an n-container

In this paper we propose a simple recursive uniform algorithm for the problem of packing n-dimensional boxes into an n-container. We are particularly concerned about the special case n=3 where the boxes can be packed in a given subset of their six possible positionings. Our method studies symmetries in the packings by the use of an ordered set of three directed graphs with the same edges (a 3-tet or triad) and induced smaller structures of the same kind named minors. With the method, degeneracy and symmetry issues, which curtail the implicit enumeration to practically acceptable running times, become transparent. In order to illustrate the performance of the algorithm, computational results from solving randomly generated 3-D examples are presented and compared with the ones of a layers and knapsack approach. The present study has real world applications for the problems of pallet and container loading.     

Single machine scheduling problems with general position-dependent processing times and past-sequence-dependent delivery times

This paper considers single machine scheduling with past-sequence-dependent (psd) delivery times, in which the processing time of a job depends on its position in a sequence. We provide a unified model for solving single machine scheduling problems with psd delivery times. We first show how this unified model can be useful in solving scheduling problems with due date assignment considerations. We analyze the problem with four different due date assignment methods, the objective function includes costs for earliness, tardiness and due date assignment. We then consider scheduling problems which do not involve due date assignment decisions. The objective function is to minimize makespan, total completion time and total absolute variation in completion times. We show that each of the problems can be reduced to a special case of our unified model and solved in O(n ³) time. In addition, we also show that each of the problems can be solved in O(nlogn) time for the spacial case with job-independent positional function.     

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.     

Efficient computation of 2-medians in a tree network with positive/negative weights

We consider a variant of the classical two median facility location problem on a tree in which vertices are allowed to have positive or negative weights. This problem was proposed by Burkard et al. in 2000 (R.E. Burkard, E. Çela, H. Dollani, 2-medians in trees with pos/neg-weights, Discrete Appl. Math. 105 (2000) 51-71). who looked at two objectives, finding the total minimum weighted distance (MWD) and the total weighted minimum distance (WMD). Their approach finds an optimal solution in O(n²) time and O(n³) time, respectively, with better performance for special trees such as paths and stars. We propose here an O(nlogn) algorithm for the MWD problem on trees of arbitrary shape. We also briefly discuss the WMD case and argue that it can be solved in time. However, a systematic exposition of the later case cannot be contained in this paper.     

Solving the bi-objective multi-dimensional knapsack problem exploiting the concept of core

This paper deals with the bi-objective multi-dimensional knapsack problem. We propose the adaptation of the core concept that is effectively used in single-objective multi-dimensional knapsack problems. The main idea of the core concept is based on the “divide and conquer” principle. Namely, instead of solving one problem with n variables we solve several sub-problems with a fraction of n variables (core variables). The quality of the obtained solution can be adjusted according to the size of the core and there is always a trade off between the solution time and the quality of solution. In the specific study we define the core problem for the multi-objective multi-dimensional knapsack problem. After defining the core we solve the bi-objective integer programming that comprises only the core variables using the Multicriteria Branch and Bound algorithm that can generate the complete Pareto set in small and medium size multi-objective integer programming problems. A small example is used to illustrate the method while computational and economy issues are also discussed. Computational experiments are also presented using available or appropriately modified benchmarks in order to examine the quality of Pareto set approximation with respect to the solution time. Extensions to the general multi-objective case as well as to the computation of the exact solution are also mentioned.     

The just-in-time scheduling problem in a flow-shop scheduling system

We study the problem of maximizing the weighted number of just-in-time (JIT) jobs in a flow-shop scheduling system under four different scenarios. The first scenario is where the flow-shop includes only two machines and all the jobs have the same gain for being completed JIT. For this scenario, we provide an O(n³) time optimization algorithm which is faster than the best known algorithm in the literature. The second scenario is where the job processing times are machine-independent. For this scenario, the scheduling system is commonly referred to as a proportionate flow-shop. We show that in this case, the problem of maximizing the weighted number of JIT jobs is NP-hard in the ordinary sense for any arbitrary number of machines. Moreover, we provide a fully polynomial time approximation scheme (FPTAS) for its solution and a polynomial time algorithm to solve the special case for which all the jobs have the same gain for being completed JIT. The third scenario is where a set of identical jobs is to be produced for different customers. For this scenario, we provide an O(n³) time optimization algorithm which is independent of the number of machines. We also show that the time complexity can be reduced to O(n log n) if all the jobs have the same gain for being completed JIT. In the last scenario, we study the JIT scheduling problem on m machines with a no-wait restriction and provide an O(mn²) time optimization algorithm.     

On some algorithmic problems regarding the hairpin completion

We consider a few algorithmic problems regarding the hairpin completion. The first problem we consider is the membership problem of the hairpin and iterated hairpin completion of a language. We propose an O(nf(n)) and O(n²f(n)) time recognition algorithm for the hairpin completion and iterated hairpin completion, respectively, of a language recognizable in O(f(n)) time. We show that the n factor is not needed in the case of hairpin completion of regular and context-free languages. The n² factor is not needed in the case of iterated hairpin completion of context-free languages, but it is reduced to n in the case of iterated hairpin completion of regular languages. We then define the hairpin completion distance between two words and present a cubic time algorithm for computing this distance. A linear time algorithm for deciding whether or not the hairpin completion distance with respect to a given word is connected is also discussed. Finally, we give a short list of open problems which appear attractive to us.     

A due-window assignment problem with position-dependent processing times

We extend a classical single-machine due-window assignment problem to the case of position-dependent processing times. In addition to the standard job scheduling decisions, one has to assign a time interval (due-window), such that jobs completed within this interval are assumed to be on time and not penalized. The cost components are: total earliness, total tardiness and due-window location and size. We introduce an O(n³) solution algorithm, where n is the number of jobs. We also investigate several special cases, and examine numerically the sensitivity of the solution (schedule and due-window) to the different cost parameters.     

On reduction of duality gap in quadratic knapsack problems

We investigate in this paper the duality gap between quadratic knapsack problem and its Lagrangian dual or semidefinite programming relaxation. We characterize the duality gap by a distance measure from set {0, 1} ⁿ to certain polyhedral set and demonstrate that the duality gap can be reduced by an amount proportional to the square of the distance. We further discuss how to compute the distance measure via cell enumeration method and to derive the corresponding improved upper bound of the problem.     

A note on maximizing a submodular set function subject to a knapsack constraint

In this paper, we obtain an (1−e⁻¹)-approximation algorithm for maximizing a nondecreasing submodular set function subject to a knapsack constraint. This algorithm requires O(n⁵) function value computations.     

Minimizing weighted earliness–tardiness and due-date cost with unit processing-time jobs

The classical single-machine scheduling and due-date assignment problem of Panwalker et al. [Panwalker, S.S., Smith, M.L., Seidmann, A., 1982. Common due date assignment to minimize total penalty for the one machine scheduling problem. Operations Research 30(2) (1982) 391–399] is the following: All n jobs share a common due-date, which is to be determined. Jobs completed prior to or after the due-date are penalized according to a cost function which is linear and job-independent. The objective is to minimize the total earliness–tardiness and due-date cost. We study a generalized version of this problem in which: (i) the earliness and tardiness costs are allowed to be job dependent and asymmetric and (ii) jobs are processed on parallel identical machines. We focus on the case of unit processing-time jobs. The problem is shown to be solved in polynomial (O(n⁴)) time. Then we study the special case with no due-date cost (a classical problem known in the literature as TWET). We introduce an O(n³) solution for this case. Finally, we study the minmax version of the problem, (i.e., the objective is to minimize the largest cost incurred by any of the jobs), which is shown to be solved in polynomial time as well.     

The oscillating random walk

{Y_n;n=0, 1, …} denotes a stationary Markov chain taking values in R^d. As long as the process stays on the same side of a fixed hyperplane E⁰, it behaves as an ordinary random walk with jump measure μ or ν, respectively. Thus ordinary random walk would be the special case μ = ν. Also the process Y′_n = |Y′_n?1?Z_n| (with the Z_n as i.i.d. real random varia bles) may be regarded as a special case. The general process is studied by a Wiener–Hopf type method. Exact formulae are obtained for many quantities of interest. For the special case that the Y_n are integral-valued, renewal type conditions are established which are necessary and sufficient for recurrence.     

Parallel solution of certain Toeplitz least-squares problems

We describe a systolic algorithm for solving a Toeplitz least-squares problem of special form. Such problems arise, for example, when Volterra convolution equations of the first kind are solved by regularization. The systolic algorithm is based on a sequential algorithm of Eldén, but we show how the storage requirements of Eldén's algorithm can be reduced from O(n²) to O(n). The sequential algorithm takes time O(n²); the systolic algorithm takes time O(n) using a linear systolic array of O(n) cells. We also show how large problems may be decomposed and solved on a small systolic array.     

A Multi-Period Renewal equipment problem

This paper looks at a Multi-Period Renewal equipment problem (MPR). It is inspired by a specific real-life situation where a set of hardware items is to be managed and their replacement dates determined, given a budget over a time horizon comprising a set of periods. The particular characteristic of this problem is the possibility of carrying forward any unused budget from one period to the next, which corresponds to the multi-periodicity aspect in the model. We begin with the industrial context and deduce the corresponding knapsack model that is the subject of this paper. Links to certain variants of the knapsack problem are next examined. We provide a study of complexity of the problem, for some of its special cases, and for its continuous relaxation. In particular, it is established that its continuous relaxation and a special case can be solved in (strongly) polynomial time, that three other special cases can be solved in pseudo-polynomial time, while the problem itself is strongly NP-hard when the number of periods is unbounded. Next, two heuristics are proposed for solving the MPR problem. Experimental results and comparisons with the Martello&Toth and Dantzig heuristics, adapted to our problem, are provided.     

Single machine scheduling with multiple common due date assignment and aging effect under a deteriorating maintenance activity consideration

In this paper, we consider the multiple common due date assignment and single machine scheduling with a job-dependent aging effect and a deteriorating maintenance activity. Once the maintenance activity has been completed, the machine will revert to its initial condition and the aging effect will start anew, the maintenance duration depends on its starting time. The objective is to minimize the total of earliness, tardiness, due date costs and find the optimal due date, the optimal maintenance position. We introduce an efficient O(n ⁴) algorithm to solve the problem. We also provide a special case of the problem and show that it remains polynomial time solvable.