branch and bound algorithm
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Algorithms ◽  
2021 ◽  
Vol 14 (12) ◽  
pp. 365
Slim Belhaiza

When several Nash equilibria exist in the game, decision-makers need to refine their choices based on some refinement concepts. To this aim, the notion of a ϵ-proper equilibria set for polymatrix games is used to develop 0–1 mixed linear programs and compute ϵ-proper Nash equilibria. A Branch-and-Bound exact arithmetics algorithm is proposed. Experimental results are provided on polymatrix games randomly generated with different sizes and densities.

OR Spectrum ◽  
2021 ◽  
Kai Watermeyer ◽  
Jürgen Zimmermann

AbstractThe concept of partially renewable resources provides a general modeling framework that can be used for a wide range of different real-life applications. In this paper, we consider a resource-constrained project duration problem with partially renewable resources, where the temporal constraints between the activities are given by minimum and maximum time lags. We present a new branch-and-bound algorithm for this problem, which is based on a stepwise decomposition of the possible resource consumptions by the activities of the project. It is shown that the new approach results in a polynomially bounded depth of the enumeration tree, which is obtained by kind of a binary search. In a comprehensive experimental performance analysis, we compare our exact solution procedure with all branch-and-bound algorithms and state-of-the-art heuristics from the literature on different benchmark sets. The results of the performance study reveal that our branch-and-bound algorithm clearly outperforms all exact solution procedures. Furthermore, it is shown that our new approach dominates the state-of-the-art heuristics on well known benchmark instances.

2021 ◽  
Vol 2021 ◽  
pp. 1-10
Pujun Jia ◽  
Hongwei Jiao ◽  
Dongwei Shi ◽  
Jingben Yin

This paper presents an efficient outer space branch-and-bound algorithm for globally solving a minimax linear fractional programming problem (MLFP), which has a wide range of applications in data envelopment analysis, engineering optimization, management optimization, and so on. In this algorithm, by introducing auxiliary variables, we first equivalently transform the problem (MLFP) into the problem (EP). By using a new linear relaxation technique, the problem (EP) is reduced to a sequence of linear relaxation problems over the outer space rectangle, which provides the valid lower bound for the optimal value of the problem (EP). Based on the outer space branch-and-bound search and the linear relaxation problem, an outer space branch-and-bound algorithm is constructed for globally solving the problem (MLFP). In addition, the convergence and complexity of the presented algorithm are given. Finally, numerical experimental results demonstrate the feasibility and efficiency of the proposed algorithm.

Mathematics ◽  
2021 ◽  
Vol 9 (23) ◽  
pp. 3045
Emili Vizuete-Luciano ◽  
Sefa Boria-Reverter ◽  
José M. Merigó-Lindahl ◽  
Anna Maria Gil-Lafuente ◽  
Maria Luisa Solé-Moro

The ordered weighted averaging (OWA) operator is one of the most used techniques in the operator’s aggregation procedure. This paper proposes a new assignment algorithm by using the OWA operator and different extensions of it in the Branch-and-bound algorithm. The process is based on the use of the ordered weighted average distance operator (OWAD) and the induced OWAD operator (IOWAD). We present it as the Branch-and-bound algorithm with the OWAD operator (BBAOWAD) and the Branch-and-bound algorithm with the IOWAD operator (BBAIOWAD). The main advantage of this approach is that we can obtain more detailed information by obtaining a parameterized family of aggregation operators. The application of the new algorithm is developed in a consumer decision-making model in the city of Barcelona regarding the selection of groceries by districts that best suit their needs. We rely on the opinion of local commerce experts in the city. The key advantage of this approach is that we can consider different sources of information independent of each other.

2021 ◽  
Vol 5 (1) ◽  
pp. 14-18
Nintia Litano Buyung ◽  
Endang Suhendar

AbstractIn maximizing the profits to be obtained the company needs optimal production planning. The plan considers the resources of the company. PT XYZ is a furniture company. This research focuses on optimizing production planning on the manufacture of door products at PT. XYZ. There are several types of products issued in: D1 type door, D2 type door, D3 type door, and D4 type door. Production planning at PT. XYZ can be seen as an integer program model, which is a method related to optimizing resources to increase profits. Optimization is done by determining the amount of production for each type and each calculating existing resources. The solution search for this model is done by the Branch and Bound algorithm. Based on the calculation results using QM software for Windows, the amount corresponding to production is obtained by using Branches and Bound giving an increase of 36.5% compared to the acquisition of PT. XYZ before. Keywords: Branch and Bound Algorithms, Integer Programming,Optimization  AbstrakDalam memaksimalkan keuntungan yang akan diperoleh perusahaan perlu adanya perencanaan produksi yang optimal. Perencanaan tersebut mempertimbangkan ketersediaan sumber daya pada perusahaan. PT XYZ merupakan perusahaan yang bergerak di bidang furniture. Penelitian ini fokus kepada pengoptimalan perencanaan produksi pada pembuatan produk pintu di PT.XYZ. Terdapat beberapa jenis produk yang diproduksi di antaranya: Pintu tipe D1, Pintu tipe D2, Pintu tipe D3, dan Pintu tipe D4. Perencanaan produksi di PT.XYZ ini dapat dikatakan sebagai model program integer, karena semua variabel menghendaki hasilnya berupa bilangan bulat. Program tersebut berhubungan dengan pengoptimalan ketersediaan sumber daya bertujuan untuk memaksimalkan keuntungan. Pengoptimalan yang dilakukan yaitu dengan menentukan jumlah produksi untuk masing-masing tipe serta mempertimbangkan semua ketersediaan sumber daya yang ada. Pencarian solusi untuk model ini dilakukan dengan algoritma Branch and Bound. Berdasarkan hasil perhitungan menggunakan software QM for Windows, diketahui bahwa penentuan jumlah produksi dengan menggunakan algoritma Branch and Bound memberikan peningkatan keuntungan sebesar 36.5% dibandingkan dengan keuntungan PT.XYZ sebelumnya. Kata kunci: Optimasi, program integer, algoritma Branch and BoundReferensi[1]     Sofyan Assauri. Manajemen Produksi dan Operasi. Lembaga Penerbit FakultasEkonomi Universitas Indonesia. Jakarta. 2008.[2]      Winston, W. L. Operations Research: Applications and Algorithms. Edisi Keempat.Canada: Brooks/Cole-Thomson Learning. 2004.[3]      Akram, S. A., dan Jaya, A. I. Optimalisasi Produksi Roti dengan Menggunakan Metode Branch and Bound (Studi Kasus Pada Pabrik Roti Syariah Bakery, Jl. Maleo, Lrg.VIII No. 68 Palu). Jurnal Ilmiah Matematika dan Terapan, 13(2): 98-107. 2016.[4]      Jiao, H. W., dkk. An Effective Branch and Bound Algorithm for MinimaxLinear Fractional Programming. Journal of Applied Mathematics, Volume 2014: 8. 2014.[5]      Williams, H. P. The Problem with Integer Programming. Journal of Management Mathematics, 22(3): 213-230. 2011.[6]      Falani, I. Penentuan Nilai Parameter Metode Exponential Smoothing dengan Algoritma Genetik dalam Meningkatkan Akurasi Forecasting. Journal of Computer Engineering System and Science, 3(1): 14–16. 2018.[7]      Mehdizadeh, E., dan Jalili, S. An Algorithm Based on Theory of Constraints and Branch and Bound for Solving Integrated Product-Mix-Outsourcing Problem. Journal of Optimization in Industrial Engineering, 12(1): 167-172. 2019.[8]      Taylor, B. W. Introduction to Management Science. Edisi ke-11. United States of America: Prentice-Hall International, INC. 2013[9]      Puryani., dan Ristono, A. Penelitian Operasional. Yogyakarta: Graha Ilmu. 2012.[10]    Yusrah N. dkk. Implementasi Algoritma Branch and Bound Dalam Penentuan Jumlah Produksi Untuk Memaksimalkan Keuntungan. Jurnal String Vol. 3 No. 1 Agustus 2018. ISSN: 2527-9661[11]    Taha, H. A. Operations Research: An Introduction. Edisi ke-8. United States of America: Prentice-Hall International, INC. 2007.

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