340 research outputs found

    Polynomial Time Algorithms For Some Multi-Level Lot-Sizing Problems With Production Capacities

    Get PDF
    We consider a model for a serial supply chain in which production, inventory, and transportation decisions are integrated, in the presence of production capacities and for different transportation cost functions. The model we study is a generalization of the traditional single-item economic lot-sizing model, adding stationary production capacities at the manufacturer, as well as multiple intermediate storage levels (including the retailer level), and transportation between these levels. Allowing for general concave production costs and linear holding costs, we provide polynomialtime algorithms for the cases where the transportation costs are either linear, or are concave with a fixed-charge structure. In the latter case, we make the additional common and reasonable assumption that the variable transportation and inventory costs are such that holding inventories at higher levels in the supply chain is more attractive from a variable cost perspective. The running times of the algorithms are remarkably insensitive to the number of levels in the supply chain

    Computational results for Constrained Minimum Spanning Trees in Flow Networks

    Get PDF
    In this work, we address the problem of finding a minimum cost spanning tree on a single source flow network. The tree must span all vertices in the given network and satisfy customer demands at a minimum cost. The total cost is given by the summation of the arc setup costs and of the nonlinear flow routing costs over all used arcs. Furthermore, we restrict the trees of interest by imposing a maximum number of arcs on the longest arc emanating from the single source vertex. We propose a dynamic programming model an solution procedure to solve this problem exactly. Intensive computational experiments were performed using randomly generated test problems and the results obtained are reported. From them we can conclude that the method performance is independent of the type of cost functions considered and improves with the tightness of the constrains.Dynamic programming, network flows, constrained trees, general nonlinear costs

    Separable Concave Optimization Approximately Equals Piecewise-Linear Optimization

    Get PDF
    We study the problem of minimizing a nonnegative separable concave function over a compact feasible set. We approximate this problem to within a factor of 1+epsilon by a piecewise-linear minimization problem over the same feasible set. Our main result is that when the feasible set is a polyhedron, the number of resulting pieces is polynomial in the input size of the polyhedron and linear in 1/epsilon. For many practical concave cost problems, the resulting piecewise-linear cost problem can be formulated as a well-studied discrete optimization problem. As a result, a variety of polynomial-time exact algorithms, approximation algorithms, and polynomial-time heuristics for discrete optimization problems immediately yield fully polynomial-time approximation schemes, approximation algorithms, and polynomial-time heuristics for the corresponding concave cost problems. We illustrate our approach on two problems. For the concave cost multicommodity flow problem, we devise a new heuristic and study its performance using computational experiments. We are able to approximately solve significantly larger test instances than previously possible, and obtain solutions on average within 4.27% of optimality. For the concave cost facility location problem, we obtain a new 1.4991+epsilon approximation algorithm.Comment: Full pape

    Models and Methods for Merge-In-Transit Operations

    Get PDF
    We develop integer programming formulations and solution methods for addressing operational issues in merge-in-transit distribution systems. The models account for various complex problem features including the integration of inventory and transportation decisions, the dynamic and multimodal components of the application, and the non-convex piecewise linear structure of the cost functions. To accurately model the cost functions, we introduce disaggregation techniques that allow us to derive a hierarchy of linear programming relaxations. To solve these relaxations, we propose a cutting-plane procedure that combines constraint and variable generation with rounding and branch-and-bound heuristics. We demonstrate the effectiveness of this approach on a large set of test problems with instances with up to almost 500,000 integer variables derived from actual data from the computer industry. Key words : Merge-in-transit distribution systems, logistics, transportation, integer programming, disaggregation, cutting-plane method

    ์šฉ๋Ÿ‰ ์ œ์•ฝ์ด ์—†๋Š” ๋ถ€๋ณด์ƒ ๋ฌธ์ œ์˜ ํ˜ผํ•ฉ ์ด์ง„ ์ด์ฐจ ๋ฌธ์ œ๋กœ์˜ ๋ชจํ˜•ํ™”๋ฅผ ํ†ตํ•œ ํ•ด๋ฒ•

    Get PDF
    ํ•™์œ„๋…ผ๋ฌธ(์„์‚ฌ) -- ์„œ์šธ๋Œ€ํ•™๊ต๋Œ€ํ•™์› : ๊ณต๊ณผ๋Œ€ํ•™ ์‚ฐ์—…๊ณตํ•™๊ณผ, 2022. 8. ํ™์„ฑํ•„.๋ถ€๋ณด์ƒ ๋ฌธ์ œ๋Š” ๋น„์ˆœํ™˜ ์œ ํ–ฅ ๊ทธ๋ž˜ํ”„ ์ƒ์—์„œ ์ถœ๋ฐœ, ๋„์ฐฉ ๋งˆ๋””๋ฅผ ์ž‡๋Š” ๊ฒฝ๋กœ์™€ ๊ทธ ๊ฒฝ๋กœ ์ƒ์˜ ํ๋ฆ„์„ ๊ฒฐ์ •ํ•˜๋Š” ๋ฌธ์ œ์ด๋‹ค. ๋ถ€๋ณด์ƒ์€ ๋„์‹œ 1์—์„œ n๊นŒ์ง€ ์ด๋™ํ•˜๋ฉด์„œ ๊ฐ ๋„์‹œ๋ฅผ ๊ฒฝ์œ ํ•˜ ๊ฑฐ๋‚˜ ์ง€๋‚˜์น˜๋ฉฐ, ๊ฒฝ์œ ํ•˜๋Š” ๋„์‹œ์—์„œ๋งŒ ์ƒํ’ˆ์„ ๋งค๋งคํ•  ์ˆ˜ ์žˆ์ง€๋งŒ ์ด๋™ ๊ฑฐ๋ฆฌ์™€ ์ƒํ’ˆ๋Ÿ‰์— ๋”ฐ๋ฅธ ๋น„์šฉ ๋˜ํ•œ ์ง€๋ถˆํ•ด์•ผ ํ•œ๋‹ค. ์ด ๋•Œ, ๋ถ€๋ณด์ƒ์€ ์ž์‹ ์˜ ์ˆ˜์ต, ์ฆ‰ ์ด ์ƒํ’ˆ์˜ ํŒ๋งค๋Ÿ‰์—์„œ ์–ป๋Š” ์ˆ˜์ต๊ณผ ์ง€๋ถˆ ๋น„์šฉ์˜ ์ฐจ๋ฅผ ์ตœ๋Œ€ํ™”ํ•˜๊ณ ์ž ํ•œ๋‹ค. ๋ณธ ์—ฐ๊ตฌ์—์„œ๋Š” ์šฉ๋Ÿ‰ ์ œ์•ฝ์ด ์—†๋Š” ๊ฒฝ์šฐ๋งŒ์„ ๋‹ค๋ฃจ๋ฉฐ ๊ธฐ์กด ๋ถ€๋ณด์ƒ ๋ฌธ์ œ๋ฅผ ํ˜ผํ•ฉ์ด์ง„์ด์ฐจ๋ฌธ์ œ์œผ๋กœ ์žฌ๋ชจํ˜•ํ™”ํ•˜์—ฌ ๋ถ„์ง€์ ˆ๋‹จ๋ฒ• ์œผ๋กœ ๋ฌธ์ œ๋ฅผ ํ‘ผ๋‹ค. ์ด ๋•Œ ๋ชฉ์  ํ•จ์ˆ˜๋ฅผ ๋ณผ๋กํ™”ํ•˜๊ณ  ์—ฐ์† ์™„ํ™”์‹œ์ผœ ์–ป์„ ์ˆ˜ ์žˆ๋Š” ์ƒํ•œ์„ ๋น„๊ตํ•˜๊ธฐ ์œ„ํ•ด ์—ฌ๋Ÿฌ ๋ณผ๋กํ™” ๋ฐฉ๋ฒ•๋“ค์„ ๋น„๊ตํ•˜๊ณ  ๋น„๊ต์‹คํ—˜ํ•œ ๊ฒฐ๊ณผ ๋˜ํ•œ ์ œ์‹œํ•œ๋‹ค.Bubosang Problem is a problem set on a directed acyclic graph path concerning both the path and multi-commodity flow decisions. A merchant travels from city 1 through n, either transiting through a city and trading products or passing by the city to the next city on his route. He wants to choose the path and trading product quantity to maximize his net profit which is defined by the difference between the total sales revenue and the traveling cost. The scope of the study considers only the uncapacitated case. In this study, we reformulate BP into a mixed binary quadratic problem to employ the branch-and-cut algorithm to solve the problem. Specifically, we compare the upper bound obtained through the continuous relaxation and convexification of the objective by studying different convexification methods. Computational results of the comparison are also provided.Chapter 1 Introduction 1 1.1 Background 1 1.2 Literature Review 3 1.3 Research Motivations 5 1.4 Organization of the Thesis 6 Chapter 2 Problem Definition and Mathematical Formulations 7 2.1 Problem Definition 7 2.2 Flow Arc Formulation 8 2.3 MBQP Formulation 11 2.3.1 MBQP 13 2.4 Branch-and-Cut Algorithm 14 2.4.1 Overall Setting 14 2.4.2 Cutset Inequality 14 2.4.3 Lower Bound 15 2.4.4 Upper Bound 18 Chapter 3 Convexification Methods 19 3.1 One Coefficient Case : Eigenvalue Method 21 3.2 Criteria for Convexification Evaluation 22 3.2.1 Criterion for Unweighted Methods 22 3.3 Two Coefficient Case : (ฮฑ, ฮฒ) - SDP method 23 3.4 Two Coefficient Case : (ฮฑ, ฮฒ) - Sum of Squares Method 24 3.5 Four Coefficient Case : (ฮฑ, ฮฒ, ฮณ, ฮด) - method 26 3.5.1 (ฮฑ, ฮฒ, ฮณ, ฮด) - SDP method 26 3.5.2 (ฮฑ, ฮฒ, ฮณ, ฮด) - Sum of Squares method 28 3.6 Five Coefficient Case : (ฮฑ, ฮฒ, ฮณ, ฮด, ฯ„ ) - Sum of Squares method 29 3.7 Weighted methods 30 3.7.1 Criterion for Weighted Methods 30 Chapter 4 Computational Experiments 32 Chapter 5 Conclusion 36 Bibliography 37 ๊ตญ๋ฌธ์ดˆ๋ก 41์„

    Multilevel Lot-Sizing with Inventory Bounds

    Get PDF
    We consider a single-item multilevel lot-sizing problem with a serial structure where one of the levels has an inventory capacity (the bottleneck level). We propose a novel dynamic programming algorithm combining Zangwillโ€™s approach for the uncapacitated problem and the basis-path approach for the production capacitated problem. Under reasonable assumptions on the cost parameters the time complexity of the algorithm is O(LT6) with L the number of levels in the supply chain and T the length of the planning horizon. Computational tests show that our algorithm is significantly faster than the commercial solver CPLEX applied to a standard formulation and can solve reasonably sized instances up to 48 periods and 12 levels in a few minutes.</p
    • โ€ฆ
    corecore