scholarly journals Dynamic Programming and Heuristic for Stochastic Uncapacitated Lot-Sizing Problems with Incremental Quantity Discount

2012 ◽  
Vol 2012 ◽  
pp. 1-21 ◽  
Author(s):  
Yuli Zhang ◽  
Shiji Song ◽  
Cheng Wu ◽  
Wenjun Yin
Keyword(s):  
Dynamic Programming ◽  
Lot Sizing ◽  
Dynamic Programming Algorithm ◽  
Optimal Solution ◽  
Mixed Integer ◽  
Programming Algorithm ◽  
Quantity Discount ◽  
Path Property ◽  
Commercial Solver

The stochastic uncapacitated lot-sizing problems with incremental quantity discount have been studied in this paper. First, a multistage stochastic mixed integer model is established by the scenario analysis approach and an equivalent reformulation is obtained through proper relaxation under the decreasing unit order price assumption. The proposed reformulation allows us to extend the production-path property to this framework, and furthermore we provide a more accurate characterization of the optimal solution. Then, a backward dynamic programming algorithm is developed to obtain the optimal solution and considering its exponential computation complexity in term of time stages, we design a new rolling horizon heuristic based on the proposed property. Comparisons with the commercial solver CPLEX and other heuristics indicate better performance of our proposed algorithms in both quality of solution and run time.

scholarly journals Lot-sizing problem with several production centers

2005 ◽  
Vol 25 (3) ◽  
pp. 479-492 ◽  
Author(s):  
Franklina Maria Bragion de Toledo ◽  
André Luís Shiguemoto
Keyword(s):  
Dynamic Programming ◽  
Lot Sizing ◽  
Dynamic Programming Algorithm ◽  
Production Capacity ◽  
Efficient Implementation ◽  
Programming Algorithm ◽  
Future Research ◽  
Production Environment ◽  
Lot Sizing Problem ◽  
Forward Dynamic Programming

In this paper, a case study is carried out concerning the lot-sizing problem involving a single item production planning in several production centers that do not present capacity constraints. Demand can be met with backlogging or not. This problem results from simplifying practical problems, such as the material requirement planning (MRP) system and also lot-sizing problems with multiple items and limited production capacity. First we propose an efficient implementation of a forward dynamic programming algorithm for problems with one single production center. Although this does not reduce its complexity, it has shown to be rather effective, according to computational tests. Next, we studied the problem with a production environment composed of several production centers. For this problem two algorithms are implemented, the first one is an extension of the dynamic programming algorithm for one production center and the second one is an efficient implementation of the first algorithm. Their efficiency are shown by computational testing of the algorithms and proposals for future research are presented.

scholarly journals A Hybrid Dynamic Programming for Solving Fixed Cost Transportation with Discounted Mechanism

2016 ◽  
Vol 2016 ◽  
pp. 1-9
Author(s):  
Farhad Ghassemi Tari
Keyword(s):  
Dynamic Programming ◽  
Programming Model ◽  
Transportation Network ◽  
Dynamic Programming Algorithm ◽  
Optimal Solution ◽  
Transportation Cost ◽  
Solution Algorithm ◽  
Transportation Costs ◽  
Programming Algorithm ◽  
Fixed Cost

The problem of allocating different types of vehicles for transporting a set of products from a manufacturer to its depots/cross docks, in an existing transportation network, to minimize the total transportation costs, is considered. The distribution network involves a heterogeneous fleet of vehicles, with a variable transportation cost and a fixed cost in which a discount mechanism is applied on the fixed part of the transportation costs. It is assumed that the number of available vehicles is limited for some types. A mathematical programming model in the form of the discrete nonlinear optimization model is proposed. A hybrid dynamic programming algorithm is developed for finding the optimal solution. To increase the computational efficiency of the solution algorithm, several concepts and routines, such as the imbedded state routine, surrogate constraint concept, and bounding schemes, are incorporated in the dynamic programming algorithm. A real world case problem is selected and solved by the proposed solution algorithm, and the optimal solution is obtained.

scholarly journals Comparison of Dynamic Programming Algorithm and Greedy Algorithm on Integer Knapsack Problem in Freight Transportation

2018 ◽  
Vol 5 (1) ◽  
pp. 49 ◽  
Author(s):  
Global Ilham Sampurno ◽  
Endang Sugiharti ◽  
Alamsyah Alamsyah
Keyword(s):  
Dynamic Programming ◽  
Greedy Algorithm ◽  
Knapsack Problem ◽  
Dynamic Programming Algorithm ◽  
Optimal Solution ◽  
Freight Transportation ◽  
Total Weight ◽  
Knapsack Problems ◽  
Programming Algorithm ◽  
Selection Of

At this time the delivery of goods to be familiar because the use of delivery of goods services greatly facilitate customers. PT Post Indonesia is one of the delivery of goods. On the delivery of goods, we often encounter the selection of goods which entered first into the transportation and  held from the delivery. At the time of the selection, there are Knapsack problems that require optimal selection of solutions. Knapsack is a place used as a means of storing or inserting an object. The purpose of this research is to know how to get optimal solution result in solving Integer Knapsack problem on freight transportation by using Dynamic Programming Algorithm and Greedy Algorithm at PT Post Indonesia Semarang. This also knowing the results of the implementation of Greedy Algorithm with Dynamic Programming Algorithm on Integer Knapsack problems on the selection of goods transport in PT Post Indonesia Semarang by applying on the mobile application. The results of this research are made from the results obtained by the Dynamic Programming Algorithm with total weight 5022 kg in 7 days. While the calculation result obtained by Greedy Algorithm, that is total weight of delivery equal to 4496 kg in 7 days. It can be concluded that the calculation results obtained by Dynamic Programming Algorithm in 7 days has a total weight of 526 kg is greater when compared with Greedy Algorithm.

scholarly journals Solving methods for the quay crane scheduling problem at port of Tripoli-Lebanon.

2020 ◽  
Author(s):  
Ali Skaf ◽  
Sid Lamrous ◽  
Zakaria Hammoudan ◽  
Marie-Ange Manier
Keyword(s):  
Dynamic Programming Algorithm ◽  
Heuristic Method ◽  
Optimal Solution ◽  
Mixed Integer ◽  
Programming Algorithm ◽  
Scheduling Problem ◽  
Exact Methods ◽  
Three Solutions ◽  
Quay Crane Scheduling ◽  
Crane Scheduling

The quay crane scheduling problem (QCSP) is a global problem and all ports around the world seek to solve it, to get an acceptable time of unloading containers from the vessels or loading containers to the vessels and therefore reducing the docking time in the terminal. This paper proposes three solutions for the QCSP in port of Tripoli-Lebanon, two exact methods which are the mixed integer linear programming and the dynamic programming algorithm, to obtain the optimal solution and one heuristic method which is the genetic algorithm, to obtain near optimal solution within an acceptable CPU time. The main objective of these methods is to minimize the unloading or the loading time of the containers and therefore reduce the waiting time of the vessels in the terminals. We tested and validated our methods for small and large random instances. Finally, we compared the results obtained with these methods for some real instances in the port of Tripoli-Lebanon.

Optimal Bus Stop Spacing Through Dynamic Programming and Geographic Modeling

2000 ◽  
Vol 1731 (1) ◽  
pp. 15-22 ◽  
Author(s):  
Peter G. Furth ◽  
Adam B. Rahbee
Keyword(s):  
Dynamic Programming ◽  
Operating Cost ◽  
Dynamic Programming Algorithm ◽  
Optimal Solution ◽  
Programming Algorithm ◽  
Demand Distribution ◽  
Bus Stop ◽  
Geographic Modeling ◽  
Bus Stop Spacing ◽  
Stop Spacing

A discrete approach was used to model the impacts of changing bus-stop spacing on a bus route. Among the impacts were delays to through riders, increased operating cost because of stopping delays, and shorter walking times perpendicular to the route. Every intersection along the route was treated as a candidate stop location. A simple geographic model was used to distribute the demand observed at existing stops to cross-streets and parallel streets in the route service area, resulting in a demand distribution that included concentrated and distributed demands. An efficient, dynamic programming algorithm was used to determine the optimal bus-stop locations. The model was compared with the continuum approach used in previous studies. A bus route in Boston was modeled, in which the optimal solution was an average stop spacing of 400 m (4 stops/mi), in sharp contrast to the existing average spacing of 200 m (8 stops/mi). The model may also be used to evaluate the impacts of adding, removing, or relocating selected stops.

scholarly journals Diversity of Solutions: An Exploration Through the Lens of Fixed-Parameter Tractability Theory

2020 ◽  
Author(s):  
Julien Baste ◽  
Michael R. Fellows ◽  
Lars Jaffke ◽  
Tomáš Masařík ◽  
Mateus de Oliveira Oliveira ◽  
...  
Keyword(s):  
Dynamic Programming ◽  
Dynamic Programming Algorithm ◽  
Practical Interest ◽  
Optimal Solution ◽  
Combinatorial Problem ◽  
Combinatorial Problems ◽  
Point Of View ◽  
Programming Algorithm ◽  
Fixed Parameter Tractability ◽  
Fixed Parameter

When modeling an application of practical relevance as an instance of a combinatorial problem X, we are often interested not merely in finding one optimal solution for that instance, but in finding a sufficiently diverse collection of good solutions. In this work we initiate a systematic study of diversity from the point of view of fixed-parameter tractability theory. We consider an intuitive notion of diversity of a collection of solutions which suits a large variety of combinatorial problems of practical interest. Our main contribution is an algorithmic framework which --automatically-- converts a tree-decomposition-based dynamic programming algorithm for a given combinatorial problem X into a dynamic programming algorithm for the diverse version of X. Surprisingly, our algorithm has a polynomial dependence on the diversity parameter.

scholarly journals An improved heuristic-dynamic programming algorithm for rectangular cutting problem

2020 ◽  
Vol 17 (3) ◽  
pp. 717-735
Author(s):  
Aihua Yin ◽  
Chong Chen ◽  
Dongping Hu ◽  
Jianghai Huang ◽  
Fan Yang
Keyword(s):  
Dynamic Programming ◽  
Dynamic Programming Algorithm ◽  
Optimal Solution ◽  
Computation Time ◽  
Programming Algorithm ◽  
Recursive Method ◽  
Two Dimensional ◽  
One Dimensional ◽  
Block Width ◽  
Cutting Problem

In this paper, the two-dimensional cutting problem with defects is discussed. The objective is to cut some rectangles in a given shape and direction without overlapping the defects from the rectangular plate and maximize some profit associated. An Improved Heuristic-Dynamic Program (IHDP) is presented to solve the problem. In this algorithm, the discrete set contains not only the solution of one-dimensional knapsack problem with small rectangular block width and height, but also the cutting positions of one unit outside four boundaries of each defect. In addition, the denormalization recursive method is used to further decompose the sub problem with defects. The algorithm computes thousands of typical instances. The computational experimental results show that IHDP obtains most of the optimal solution of these instances, and its computation time is less than that of the latest literature algorithms.

scholarly journals Global Optimization for Bus Line Timetable Setting Problem

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Qun Chen
Keyword(s):  
Dynamic Programming ◽  
Global Optimization ◽  
Dynamic Programming Algorithm ◽  
Optimal Solution ◽  
Programming Algorithm ◽  
Computational Time ◽  
Global Optimal Solution ◽  
Time Period ◽  
Flow Intensity ◽  
Global Optimal

This paper defines bus timetables setting problem during each time period divided in terms of passenger flow intensity; it is supposed that passengers evenly arrive and bus runs are set evenly; the problem is to determine bus runs assignment in each time period to minimize the total waiting time of passengers on platforms if the number of the total runs is known. For such a multistage decision problem, this paper designed a dynamic programming algorithm to solve it. Global optimization procedures using dynamic programming are developed. A numerical example about bus runs assignment optimization of a single line is given to demonstrate the efficiency of the proposed methodology, showing that optimizing buses’ departure time using dynamic programming can save computational time and find the global optimal solution.

scholarly journals The dynamic programming approach to long term production planning in agriculture

2011 ◽  
Vol 59 (2) ◽  
pp. 129-136 ◽  
Author(s):  
Jitka Janová
Keyword(s):  
Dynamic Programming ◽  
Production Planning ◽  
Programming Model ◽  
Dynamic Programming Algorithm ◽  
Optimal Solution ◽  
Goal Function ◽  
Programming Algorithm ◽  
Programming Approach ◽  
Model Case ◽  
Local Character

The production planning in agriculture is one of the most important decision problems of the farmer. Although some decision support tools based mainly on linear programming and addressed to agriculture authorities were presented, their direct application by a farmer is not possible. This is mainly due to the local character of the models developed for particular agricultural conditions and also due to the complexness of underlying mathematical programming models.This paper aims to develop dynamic programming model for the long run crop plan optimization covering the typical conditions of Czech farms, which could serve as a platform for further enlargements and changes according to needs and conditions of particular farm. The dynamic programming algorithm is developed in detail for model case of four areas to be planted by four crops each year. The possibility of covering different constraints by generating the state space is discussed, and the generating procedure for crop rotation rules is shown. The goal function reflects the farmers objective of profit maximization and it is defined with respect to harvests’ randomness. The case study is solved for the data from South Moravian agriculture cooperative and the optimal solution is presented and discussed.

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