Computational analysis of a binary tree based algorithm for the triangle scheduling problem

This paper deals with aggregate planning of Reconfigurable Assembly Lines (RAL). The assembly line considered in this paper consists of hexagonal cells. These have multiple slots where processing modules can be inserted to perform certain operations. In addition, each cell has a single central slot where a central module can be inserted for inter-cellular and intra-cellular transportation of parts. Multiple products with different assembly sequences must be handled over multiple planning periods. An Integer Quadratic Programming (IQP) model is proposed to solve the following problems simultaneously: (i) assigning processing modules and a central module to the cells; (ii) installation of the cells and conveyors between the cells; and (iii) routing products, ensuring that availability of the resources is not exceeded. The assembly line should be reconfigured over time to adapt to possible product functionality and demand changes at minimum reconfiguration, operational and material handling costs while ensuring the demand is met within each period. The IQP model is implemented and solved for an illustrative problem and its extensions using Gurobi.

In order to solve a kind of quadratic integer programming problem with unrestrained conditions and multi-objective function, improvements on the traditional branch and bound algorithm are carried out in this paper. By introducing response function, a practical branch and bound algorithm to solve the quadratic integer programming model is derived. The procedure of the proposed algorithm is explicated in this paper and an example with two manufactures is studied. Numerical results show that the proposed algorithm is more effective than the traditional branch and bound algorithm, and fit for more complex problems.

Dynamic VAR compensation (DVC) planning is important in power system planning to enhance short‐term voltage stability. However, since its optimisations models include differential algebraic equations (DAEs), existing methods cannot directly solve the problem as that in the optimal power flow problem. In this study, the authors design an integer quadratic programming (IQP) model for the DVC planning problem which can simultaneously optimise the sites and size of DVCs. On the basis of the defined dynamic correlation indices, a quadratic programming objective function is defined to minimise the total investment of DVCs as well as to avoid redundant installation of DVCs on adjacent buses. Subsequently, sensitivity coefficients about the control effect of DVCs are introduced to form the linear inequations as the substitute for DAEs in constraints. Then, an iterative process is applied to solve the IQP. The proposed method is applied on a modified NE 39‐bus system and a real power grid. Comparisons to the existing methods are carried out to show its superiority in model complexity and solving efficiency.

Alignment of protein-protein interfaces plays a key role in the studies of protein function, signal transduction network and drug design. We give a protein-protein interface alignment method which is based on an integer quadratic programming model. This alignment method integrates sequence evolutionary information and structural information of proteins, and scores the aligned residues using the similarity of their evolutionary profiles. Furthermore, we use the alignment method to identify evolutionally and structurally conserved residues. Through the computational experiments, we show that those evolutionally and structurally conserved residues are likely to be the key residues for protein-protein interaction, i.e. hot spots.

In the current study, an unrelated parallel machine scheduling problem with release dates is considered, which is to obtain a job assignment with minimal sum of weighted completion times. Although this problem is NP-hard in the strong sense, which renders the optimality seeking a formidable task within polynomial time, a 4-approximation algorithm based on the constant worst-case bound is devised and proved in comparison with the existing 16/3-approximation (Hall et al. in Math Oper Res 22(3):513–544, 1997). In the newly proposed algorithm, the original scheduling problem is divided into several sub-problems based on release dates. For each sub-problem, a convex quadratic integer programming model is constructed in accordance with the specific problem structure. Then a semi-definite programming approach is implemented to produce a lower bound via the semi-definite relaxation of each sub-problem. Furthermore, by considering the binary constraint, a branch and bound based method and a local search strategy are applied separately to locate the optimal solution of each sub-problem. Then the solution of the original scheduling problem is derived by integrating the outcomes of the sub-problems via the proposed approximation algorithm. In the numerical analysis, it is discovered that the proposed methods could always obtain a scheduling result within $$1\%$$ of the optimal solution. And an asymptotic trend could be observed where the quality of solutions improves even further as the scale of the scheduling problem increases.

In this paper we study the problem of minimizing weighted earliness and tardiness on a single machine when all the jobs share the same due date. We propose two quadratic integer programming models for solving both cases of unrestricted and restricted due dates, an auxiliary model based on unconstrained quadratic integer programming and an algorithmic scheme for solving each instance, according to its size and characteristics, in the most efficient way. The scheme is tested on a set of well-known test problems. By combining the solutions of the three models we prove the optimality of the solutions obtained for most of the problems. For large instances, although optimality cannot be proved, we actually obtain optimal solutions for all the tested instances.

Software Defined Networks (SDN) emerged as a new network paradigm to address the customization and flexibility problems in traditional computer networks. In SDNs, a logical centralized programmable controller manages the whole networks by installing rules onto switches. It is widely regarded that one controller is restricted on both performance and scalability. To address these limitations, pioneering researchers advocate deploying multiple controllers in SDNs where each controller is in charge of a set of switches. This raises the switch-controller association problem on one switch shall be managed by which controller. In this paper, we specially investigate minimum cost switch-controller association (MC-SCA) problem on how to minimize the number of controllers needed in an SDN while guaranteeing the flow setup time. A quadratic integer programming model is first proposed and then transformed into an equivalent integer linear programming model to describe the MC-SCA problem, which is then proved as NP-Hard. We further propose a heuristic algorithm and extensively prove its high efficiency via simulations.

Cast planning is a practical problem frequently encountered in steel industry. Its task is to group charges into batches (casts) with respect to the similarities of steel-grade and dimensions between charges, taking account of the practical technique constraints on life-span of tundish. Effective cast planning can reduce the changeover cost of charges and enhance the productivity of continuous casters. The objective under our consideration is to minimize the total dissimilarity costs between the charges in the same casts, to minimize the number of casts and the number of unselected charges. A quadratic integer programming model with multiple objectives for this problem is formulated. It is NP-complete, and so an iterated local search (ILS) algorithm is developed for the problem. In this algorithm, cyclic transfer neighborhood is adopted, in which several charges are transferred among casts simultaneously as a manner of cycle. A new kick strategy is developed with the idea of assigning charges to different casts according to the dissimilarity costs between them and the casts’ central charges identified by the current solution. Computational results using real data from an advanced iron & steel company in China indicate that the ILS algorithm provides optimal solutions for small instances, and better near-optimal solutions for larger instances compared with a linear solver, Lingo 8.0, used to solve the equivalent linear integer programming obtained by transforming the original model. Totally, 92.3% of the instances are solved to global optima by the ILS algorithm while it is possible that the other 9.7% instances also are solved to optimal, which indicates the efficiency of the algorithm. At the same time, the algorithm also provides better solutions than the ones obtained by the current system used in the company.

The curricular design of the academic programs and the distribution of the academic load has become a controversial topic since it is related to the health and performance of the students. In Chile, the load of the academic programs has to be distributed and balanced according to the Chilean Academic Credit Transfer System recommendations. The Balanced academic curriculum problem has been studied in computer science, being defined as the assignation of courses to academic periods satisfying all the prerequisite constraints. In this work, we implement two strategies to deal with this problem, a quadratic integer programming model and a genetic algorithm approach, and evaluate them using well-know data sets from the literature. Next, these strategies are adapted to study the balance of the load of six real engineering academic programs according to the Chilean credit references. The results show that both approaches obtained the optimal solutions for the literature data sets, and were able to find solutions which improve the balance of the real academic programs, distributing uniformly the load between periods.

In this paper, we introduce a modulation code design problem where best selection of two-dimensional codewords are determined to reduce two-dimensional (2D) intersymbol interference (ISI) and interpage interference (IPI). Codeword selection problem is formulated as a quadratic integer programming model that minimizes intra- and inter-symbol differences subject to uniform symbol usage and minimal Hamming distance violations. Second integer programming model detects the occurrence of isolated pixel pattern in the selected codewords. The proposed models are applied to 4-level and 2-level 6/8 codes.

This paper investigates identification of parameter and measurement errors in the power network model. A mixed integer quadratic programming (MIQP) model is proposed in this paper to precisely identify multiple parameter and measurement errors. In the proposed MIQP model, erroneous parameters and measurements are involved with penalty factor in the aim of minimizing weighted least-squares (WLS) objective function. The proposed method has the desired property of accurate identification with avoiding over identification in the meantime. Parameter and measurement errors can also be distinguished simultaneously in MIQP model. All these features are verified via simulations that are carried out on IEEE 14-bus system.

In order to facilitate decision relation of nodes, support dynamic updates and improves the speed for XML data query, etc, this paper proposes a XML document indexing structure model based on binary tree encoding. The XML document tree uses trigeminal linked list of binary tree structure to encode nodes. The indexing model of binary sort tree was established, which uses elements of the leaf node as indexing terms, and combines with semantic information of nodes. This paper gives some corresponding algorithms, implements the prototype system of indexing model and its corresponding simulation experiments. Theoretical analysis and experimental results show that the indexing model not only supports update operation of nodes and facilitates decision relation of nodes, but also has advantage of short query time.

Extension of bivariate means to weighted means of several arguments by using binary trees

As the power grid transforms towards a renewable energy dominant grid, which are associated with frequency stability challenges, it is important to combat these challenges through proper power system planning. This paper therefore investigates the impact of hybrid energy storage (HES) in enhancing the frequency stability of the modern grid. The optimization problem is formulated as a mixed integer quadratic constrained programming model to minimize the system cost, while maximizing system inertia. The proposed model is then implemented on an IEEE 6-bus system, and solved using CPLEX solver in GAMS, with its performance evaluated using two study cases. Case 1 considers only the economic objective (cost), while case 2 considers both the economic objective (cost) and the technical objective (system inertia). The result show that the incorporation of HES in the generation planning model improved the overall system inertia of the system by 11% (from 6.199s in case 1, to 6.901s in case 2), however the total system cost increased by 3% (from 218M$ in case 1 to 2.25M$ in case 2).

Optimal operation of integrated energy system considering virtual heating energy storage