Branch And Bound Method Research Articles

OPTIMIZING CARTON PRODUCT DELIVERY BY SOLVING TRAVELLING SALESMAN PROBLEM AT PACKAGING COMPANIES

Optimization of delivery routes is one form of increasing business productivity to achieve the company's main goal of distributing each product to customers. The Traveling Salesman Problem (TSP) is a combinatorial optimization problem where a salesman goes on a distribution journey that starts from the depot, then visits all customers exactly once, and ends with returning to the depot. This study aims to optimize the distribution route of carton products for packaging companies using the TSP model. The research methodology includes observation, interviews, and document studies to understand the distribution process of carton products at packaging companies. To complete the TSP model, Branch and Bound (BB) and Nearest Neighbor (NN) methods are applied to find the best solution for determining the distribution route of carton products. The way the BB method works is by utilizing branch cuts and boundaries to reduce search space and speed up the resolution process. In the NN method, the nearest point is chosen to get the shortest route distance. Research findings show that the use of the BB method results in a mileage difference from the initial route of 125 km (53% more efficient) and a fuel cost difference of 121,231 IDR (45% cheaper). Meanwhile, the NN method results in a mileage difference from the initial route of 115 km (48.94%) and a fuel cost difference of 109,901 IDR (41.27%). So, the method that produces the best solution is the BB method. The limitations of this study lie in the scale of the model used and the assumptions underlying the analysis. Future research can broaden the scope of the model and consider other factors that may affect the distribution of carton products. The results of this study contribute to improving the efficiency of carton product distribution.

Sub‐optimal Internet of Thing devices deployment using branch and bound method

AbstractThe Internet of Thing (IoT) network deployments are widely investigated in 4G and 5G systems and will still be key technical systems to drive massive connectivity of 6G systems. In 6G, IoT systems will operate with 6G new technologies such as Integrated Sensing and Communications. The IoT systems of 6G will be a platform to collect information in real world and create new use cases and business models. As the IoT devices and cellular networks are getting smarter, the IoT ecosystem allows us to bridge between human life and digital life and accelerate the transition towards a hyper‐connected world. Optimal and scalable IoT network design has been investigated in many research groups but key challenges in this topic still remain. An IoT devices deployment problem is investigated to minimise the transmission and computation cost among network nodes. The IoT devices deployment problem is formulated as Mixed‐Integer Nonlinear Programming problem. After relaxing the constraints and transforming the problem to a mixed integer linear programming (MILP) problem, the authors propose a new branch and bound (BB) method with a machine learning function and solve the MILP problem as a sub‐optimal solution. In the numerical analysis, the authors evaluate both conventional BB method and the proposed BB method with weighting factors and compare the objective function values, the number of explored nodes, and computational time. The performances of the proposed BB method are significantly improved under the given simulation configuration. The author finds the optimal mapping of IoT devices to fusion nodes.

Perencanaan Produksi Untuk Mengoptimalkan Keuntungan Dengan Metode Branch And Bound Di UKM “X”

UKM “X” Small and Medium Enterprise is a home industry which is engaged in the food industry. Productsthat are produced include sweet potato balls, crispy sweet potatoes and chips.This industry does not have a certainmethod that is certain in determining the amount of production of each type of product that is produced and resultsin profits that are less than optimal.Therefore, research is needed to find out the optimal product formulation in themanufacturing of the three products at UKM “X”.The method to be used in this study is the Branch and Boundmethod which first calculates the value of the decision variable by using POM software.This research is reviewedbased on the amount of raw material inventory, raw material for making the three products, production data,production costs, selling prices ofeach type ofproduct, and profit. So that from the analysis ofthe branch andbound method the profit value of Rp.4,514,091, for special case in June net profit become Rp 777.589,-. So thenumber of the three products that can be produced from the available ingredients for a month is 615 packs sweetpotatoballs,80packscrispysweetpotatoesand44packschips.TheproductionplanninghasincreasedprofitsbyRp 1.181.993fromprevious averagesalesprofit.

Computationally efficient evolutionary optimisation for joint reservoir design-operations

This study contributes to emerging computationally efficient methods to account for multi-purpose reservoir operations at the dam design stage. In many cases, the water supply reliability objective introduces binary variables into an otherwise simpler linear programming (LP) formulation. This has led to widespread use of evolutionary methods such as particle swarm optimisation (PSO). Combining the branch and bound (BB) method with a hybrid PSO–LP, this study presents a new BB–PSO–LP algorithm to solve the joint design and operations problem. The proposed BB–PSO–LP algorithm was benchmarked against classical versions of the BB algorithm and PSO–LP. Superiority of the BB–PSO–LP algorithm was demonstrated by finding slightly better solutions in a shorter time. That said, PSO–LP also led to acceptable results in a similar time. Although the BB algorithm was found to be efficient in problem solving, under certain conditions it is difficult for the BB method to find a feasible solution.

Job Shop Scheduling by Branch and Bound Using Genetic Programming

A classical depth-first branch and bound (BB) method is adopted to minimize the makespan of job shops based on the disjunctive graph model. The engine of the BB is Giffler-Thompson’s active schedule generation method. The performance of the BB method highly depends on the selection of child nodes in earlier branching stages. To support the selection decision, several features of nodes are stored under the BB method, and the correct selection at each branching stage is informed by the mixed-integer linear programming model. The stored data of a test problem instance is analyzed by genetic programming (GP) to generate rules for selecting the correct nodes. The depth-first BB method guided by the generated rules by GP is applied for 42 benchmark instances and exhibits competitive performance when compared with the baseline rule that always selects the child node with the smallest lower bound on makespan.

A Fully Distributed Voltage Optimization Method for Distribution Networks Considering Integer Constraints of Step Voltage Regulators

With the increasing penetration of distributed photovoltaics (PVs), the operation and control of distribution networks (DNs), especially voltage control, have become more complicated. To deal with the voltage violation problem caused by large-scale PV access, this paper presents a fully distributed optimization method that combines the alternative direction multiplier method (ADMM) with the branch and bound method (BBM) for regional DNs. The total cost of active power losses and PV generation losses is minimized by making full use of the voltage regulation resources, e.g., reactive power compensators, step voltage regulators (SVR), and PV inverters, and the ADMM is employed to realize the intra-regional optimization and inter-regional coordination. To overcome the non-convex problem that is introduced by the SVR, the constraints of real-value tap positions are reformulated as linear inequality constraints of boundary voltages and added to the original problem, then the integer optimal solutions of SVR tap positions are obtained by BBM. The effectiveness of the proposed method is verified via numerical simulations on a practical 32-bus DN in China and a modified IEEE123-bus system.

Energy-Efficient Sensor Grouping for IEEE 802.11ah Networks With Max-Min Fairness Guarantees

For the large scale wireless networks, restricted access window (RAW) mechanism is a promising technique for realizing large-scale sensor access with the limited collision probability. In this paper, we are committed to designing the traffic distribution based sensor grouping scheme to balance the energy efficiency (EE) of different groups in the large scale access networks. Specifically, by adopting the Markov chain model, we formulate the optimization problem of max-min EE by taking into account traffic demands with even distribution of different all groups, but the formulated problem is an integer nonlinear programming (INLP) problem. In order to solve the INLP problem, we propose an optimal traffic grouping algorithm (OTGA) by utilizing the branch-and-bound method (BBM) to accommodate for the congestion level among groups. Though the traffic demands of each group can be obtained from the traffic grouping scheme, different combination of heterogeneous sensors can generate the same traffic demands, which make it difficult to find the optimal solution of sensor grouping from the proposed traffic grouping scheme. Furthermore, a heuristic traffic-sensor mapping algorithm (HTMA) is presented to make the traffic demands of each group appropriate. Thus, the proposed scheme can achieve a sub-optimal performance with the individual EE. The numerical results are provided to verify the effectiveness of the proposed schemes.

Monitoring ratio of carbon to nitrogen (C/N) in wheat and barley leaves by using spectral slope features with branch-and-bound algorithm

Ratio of carbon to nitrogen concentration (C/N) that can illuminate metabolic status of C and N in crop leaves is one valuable indicator for crop nutrient diagnosis. This study explored the feasibility of using spectral slope features from hyperspectral measurements with Branch-and-Bound (BB) algorithm to monitor leaf C/N in wheat and barley. Experimental data from barley in 2010 and wheat in 2012 were collected and used. The analyses prove that leaf C/N is closely related to leaf N concentration (LNC), which implies that it is feasible to apply spectral technology to monitor leaf C/N in that LNC may have been effectivly estimated by hyperspectral measurements. The results also show that many spectral slope features proposed in this study exhibit the significant correlations with leaf C/N. The best slope feature could evaluate changes of leaf C/N well, with R2 of 0.63 for wheat, 0.68 for barley and 0.65 for both species combined, respectively. using BB algorithm with input of optiaml four slope features can improve the accuracy of leaf C/N estimations with R2 over 0.81. It is concluded that using the spectral slope new features with BB method appears very promising and potential for remotely monitoring leaf C/N in crops.

A Branch and Bound Approach to System Identification based on Fixed-rank Hankel Matrix Optimization

We consider identification of linear systems with a certain order from a set of noisy input-output observations. We utilize the fact that the system order corresponds to the rank of the Hankel matrix associated with the system impulse response. Then, the system identification problem is formulated as the minimization of the output error subject to a rank constraint on a Hankel matrix. As this problem is non-convex, we propose a branch and bound (BB) solver, which is a powerful tool for solving non-convex problems to optimality. The main ingredients of the proposed BB method are a convex relaxation problem and a local minimizer of the original non-convex problem. We illustrate the promising performance of the proposed scheme in a system identification problem. The results demonstrate the higher accuracy and stability of our method in estimating the true system compared to the standard output error (OE) algorithm.

Energy-Efficient Downlink Resource Allocation in Heterogeneous OFDMA Networks

In this paper, we investigate energy-efficient downlink resource allocation in heterogeneous orthogonal frequency division multiple access (OFDMA) networks, and formulate the energy efficiency (EE) maximization problem as a mixed-integer nonlinear fractional programing (MINLFP) problem with a nonconcave nonlinear objective function and nonlinear constraints. By means of fractional programing and changing of variables, we transform the original MINLFP problem into an equivalent optimization problem in a parametric subtractive form, which is proved to be a concave mixed-integer nonlinear programing (MINLP) problem and is optimally solved by using Dinkelbach and branch-and-bound (BB) methods. In BB method, the concave MINLP problem is relaxed to a series of concave nonlinear programing problems and solved by the use of Powell–Hestenes–Rockafellar augmented Lagrangian (PHR-AL) method. The optimal solution can be used to benchmark the performance of suboptimal solutions. As the computational complexity of BB method increases exponentially with problem size, we further develop a suboptimal two-step scheme, which first allocates the resource blocks and then performs the transmit power control to give suboptimal solution with much lower complexity. Simulation results demonstrate the effectiveness of the proposed schemes and show that the proposed suboptimal two-step scheme is promising for practical applications as it makes a good tradeoff between EE performance and computational complexity.

Branch and bound methods based tone injection schemes for PAPR reduction of DCO-OFDM visible light communications.

In this paper, the peak-to-average power ratio (PAPR) reduction problem with tone injection (TI) in the direct current-biased optical orthogonal frequency division multiplexing (DCO-OFDM) system is investigated, which is formulated as a tough integer combinatorial optimization problem. Since it is a challenging task to find the global optimal solution, the branch-and-bound method (BBM), which is extensively employed to deal with the NP-hard problem, is introduced to solve this problem. By splitting the superior branches and pruning the inferior branches, the close optimal solution is obtained. Simulation results reveal that the proposed BBM-based TI method has superior PAPR reduction and bit error rate (BER) performance compared with some existing algorithms. The proposed algorithm ensures a relatively low peak value, and thus provides an important benchmark in performance evaluations relative to other existing algorithms targeting at the same problem.

An optimal algorithm for Global Optimization and adaptive covering

The general class of zero-order Global Optimization problems is split into subclasses according to a proposed "Complexity measure" and the computational complexity of each subclass is rigorously estimated. Then, the laboriousness (computational demand) of general Branch-and-Bound (BnB) methods is estimated for each subclass. For conventional "Cubic" BnB based on splitting an n-dimensional cube into $$2^n$$2n sub-cubes, both upper and lower laboriousness estimates are obtained. The value of the Complexity measure for a problem subclass enters linearly into all complexity and laboriousness estimates for that subclass. A new BnB method based on the lattice $$A_n^*$$Anź is presented with upper laboriousness bound that is, though conservative, smaller by a factor of $$O((4/3)^n)$$O((4/3)n) than the lower bound of the conventional method. The optimality of the new method is discussed. All results are extended to the class of Adaptive Covering problems--that is, covering of a large n-dimensional set by balls of different size, where the size of each ball is defined by a locally computed criterion.

A tool for simulating parallel branch-and-bound methods

Abstract The Branch-and-Bound method is known as one of the most powerful but very resource consuming global optimization methods. Parallel and distributed computing can efficiently cope with this issue. The major difficulty in parallel B&B method is the need for dynamic load redistribution. Therefore design and study of load balancing algorithms is a separate and very important research topic. This paper presents a tool for simulating parallel Branchand-Bound method. The simulator allows one to run load balancing algorithms with various numbers of processors, sizes of the search tree, the characteristics of the supercomputer’s interconnect thereby fostering deep study of load distribution strategies. The process of resolution of the optimization problem by B&B method is replaced by a stochastic branching process. Data exchanges are modeled using the concept of logical time. The user friendly graphical interface to the simulator provides efficient visualization and convenient performance analysis.

Maximization of Worst-Case Weighted Sum-Rate for MISO Downlink Systems With Imperfect Channel Knowledge

The problem of robust weighted sum-rate maximization (WSRMax) in multicell downlink multi-input single-output systems is considered. We assume that channel state information (CSI) of all users is imperfectly known at the base stations. The problem is known to be NP-hard even in the case of perfect CSI. We propose optimal and suboptimal but fast-converging algorithms for WSRMax problem with CSI errors. Assuming bounded ellipsoidal model for the CSI errors, we optimize the worst-case weighted sum-rate. The proposed optimal algorithm is based on branch and bound (BB) technique, and it globally solves the worst-case WSRMax problem with an optimality certificate. As the convergence speed of the BB method can be slow for large networks, we also provide a fast but possibly suboptimal algorithm based on alternating optimization technique and sequential convex programming. The optimal BB based algorithm can be used to provide performance benchmarks for any suboptimal algorithm. Numerical results show that the convergence speed of the suboptimal algorithm is fast, and it finds a close-to-optimal solution in only a few iterations.

A turning restriction design problem in urban road networks

Turning restriction is one of the commonest traffic management techniques and an effective low cost traffic improvement strategy in urban road networks. However, the literature has not paid much attention to the turning restriction design problem (TRDP), which aims to determine a set of intersections where turning restrictions should be implemented. In this paper, a bi-level programming model is proposed to formulate the TRDP. The upper level problem is to minimize the total travel cost from the viewpoint of traffic managers, and the lower level problem is to depict travelers’ route choice behavior based on stochastic user equilibrium (SUE) theory. We propose a branch and bound method (BBM), based on the sensitivity analysis algorithm (SAA), to find the optimal turning restriction strategy. A branch strategy and a bound strategy are applied to accelerate the solution process of the TRDP. The computational experiments give promising results, showing that the optimal turning restriction strategy can obviously reduce system congestion and are robust to the variations of both the dispersion parameter of the SUE problem and the level of demand.

Joint Beamforming and User Maximization Techniques for Cognitive Radio Networks Based on Branch and Bound Method

We consider a network of cognitive users (also referred to as secondary users (SUs)) coexisting and sharing the spectrum with primary users (PUs) in an underlay cognitive radio network (CRN). Specifically, we consider a CRN wherein the number of SUs requesting channel access exceeds the number of available frequency bands and spatial modes. In such a setting, we propose a joint fast optimal resource allocation and beamforming algorithm to accommodate maximum possible number of SUs while satisfying quality of service (QoS) requirement for each admitted SU, transmit power limitation at the secondary network basestation (SNBS) and interference constraints imposed by the PUs. Recognizing that the original user maximization problem is a nondeterministic polynomial-time hard (NP), we use a mixed-integer programming framework to formulate the joint user maximization and beamforming problem. Subsequently, an optimal algorithm based on branch and bound (BnB) method has been proposed. In addition, we propose a suboptimal algorithm based on BnB method to reduce the complexity of the proposed algorithm. Specifically, the suboptimal algorithm has been developed based on the first feasible solution it achieves in the fast optimal BnB method. Simulation results have been provided to compare the performance of the optimal and suboptimal algorithms.

Selection of Optimal, Controlled Variables for the TEALARC LNG Process

This article shows how controlled variables (CVs) of the regulatory control layer in a liquefied natural gas (LNG) plant can be chosen as linear combinations of measurements using self-optimizing control principles. By self-optimizing control, the CVs are chosen such that the set points of the CVs remain close to steady-state optimal despite disturbances, thus reducing the need for online reoptimization. Several methods for calculation of linear combinations within this framework are compared. Self-optimizing control design can also be used in the process design phase to place measurements by reducing a maximum candidate set of measurements to a best possible subset of measurements giving an acceptable loss. This article proposes a relatively simple method for successive selection (SS) of measurements and compares this approach to a more comprehensive branch-and-bound (BB) method for selection of measurements. The results indicate that, although the BB method gives lower average losses for very small subsets, the methods are comparable with respect to average losses for medium and large subsets, and the SS method outperforms the BB method in terms of computational load.

Branch and bound method for multiobjective pairing selection

Most of the available methods for selection of input–output pairings for decentralized control require evaluation of all alternatives to find the optimal pairings. As the number of alternatives grows rapidly with process dimensions, pairing selection through an exhaustive search can be computationally forbidding for large-scale processes. Furthermore, the different criteria can be conflicting necessitating pairing selection in a multiobjective optimization framework. In this paper, an efficient branch and bound (BAB) method for multiobjective pairing selection is proposed. The proposed BAB method is illustrated through a biobjective pairing problem using selection criteria involving the relative gain array and the μ -interaction measure. The computational efficiency of the proposed method is demonstrated by using randomly generated matrices and the large-scale case study of cross-direction control.

Specialised branch-and-bound algorithm for transmission network expansion planning

An algorithm is presented that finds the optimal plan long-term transmission for all cases studied, including relatively large and complex networks. The knowledge of optimal plans is becoming more important in the emerging competitive environment, in which the correct economic signals have to be sent to all participants. The paper presents a new specialised branch-and-bound algorithm for transmission network expansion planning. Optimality is obtained at a cost, however: that is the use of a transportation model for representing the transmission network, in this model only the Kirchhoff current law is taken into account (the second law being relaxed). The expansion problem then becomes an integer linear program (ILP) which is solved by the proposed branch-and bound method without any further approximations. To control combinatorial explosion the branch and bound algorithm is specialised using specific knowledge about the problem for both the selection of candidate problems and the selection of the next variable to be used for branching. Special constraints are also used to reduce the gap between the optimal integer solution (ILP program) and the solution obtained by relaxing the integrality constraints (LP program). Tests have been performed with small, medium and large networks available in the literature.

An innovative technique for backbone network design

An algorithm is proposed for solving the minimum-cost backbone network design problem. The method includes time delay as the performance criterion and multiple-hop communication paths. The algorithm combines a branch-and bound method with the algorithm of L.R. Ford and D.R. Fulkerson (1962). The branch-and-bound method is used to find the network topology. A depth-first search using a cost heuristic that is a linear function of the maximum and current depth in the tree is used. When a feasible topology is found, the Ford-Fulkerson algorithm is applied to find the optimal paths. The resulting network is retained if it is the current optimum cost network. The algorithm is compared with a branch-and-bound algorithm and is found to produce globally optimal solutions. For large, sparsely connected networks computation times are reduced by a factor of over one thousand.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>