A hybrid two-stage transportation and batch scheduling problem

We study the coordinated scheduling problem of hybrid batch production on a single batching machine and two-stage transportation connecting the production, where there is a crane available in the first-stage transportation that transports jobs from the warehouse to the machine and there is a vehicle available in the second-stage transportation to deliver jobs from the machine to the customer. As the job to be carried out is big and heavy in the steel industry, it is reasonable assumed that both the crane and the vehicle have unit capacity. The batching machine processes a batch of jobs simultaneously. Each batch occur a setup cost. The objective is to minimize the sum of the makespan and the total setup cost. We prove that this problem is strongly NP-hard. A polynomial time algorithm is proposed for a case where the job transportation times are identical on the crane or the vehicle. An efficient heuristic algorithm for the general problem is constructed and its tight worst-case bound is analyzed. In order to further verify the performance of the proposed heuristics, we develop a lower bound on the optimal objective function. Computational experiments show that the heuristic algorithm performs well on randomly generated problem instances.     

A column generation based decomposition algorithm for a parallel machine just-in-time scheduling problem

We propose a column generation based exact decomposition algorithm for the problem of scheduling n jobs with an unrestrictively large common due date on m identical parallel machines to minimize total weighted earliness and tardiness. We first formulate the problem as an integer program, then reformulate it, using Dantzig–Wolfe decomposition, as a set partitioning problem with side constraints. Based on this set partitioning formulation, a branch and bound exact solution algorithm is developed for the problem. In the branch and bound tree, each node is the linear relaxation problem of a set partitioning problem with side constraints. This linear relaxation problem is solved by column generation approach where columns represent partial schedules on single machines and are generated by solving two single machine subproblems. Our computational results show that this decomposition algorithm is capable of solving problems with up to 60 jobs in reasonable cpu time.     

Global optimization algorithm for sum of generalized polynomial ratios problem

In this paper, a global optimization algorithm is proposed for solving sum of generalized polynomial ratios problem (P) which arises in various practical problems. Due to its intrinsic difficulty, less work has been devoted to globally solve the problem (P). For such problems, we present a branch and bound algorithm. In this method, by utilizing exponent transformation and new three-level linear relaxation method, a sequence of linear relaxation programming of the initial nonconvex programming problem (P) are derived which are embedded in a branch and bound algorithm. The proposed method need not introduce new variables and constraints and it is convergent to the global minimum of prime problem by means of the subsequent solutions of a series of linear programming problems. Several numerical examples in the literatures are tested to demonstrate that the proposed algorithm can systematically solve these examples to find the approximate ?-global optimum.     

A note on “a dual-ascent approach to the fixed-charge capacitated network design problem”

We show by a counterexample that the dual-ascent procedure proposed by Herrmann, Ioannou, Minis and Proth in a 1996 issue of the European Journal of Operational Research is incorrect in the sense that it does not generate a valid lower bound to the optimal value of fixed-charge capacitated network design problems. We provide a correct dual-ascent procedure based on the same ideas and we give an interpretation of it in terms of a simple Lagrangean relaxation. Although correct, this procedure is not effective, as in general, it provides a less tighter bound than the linear programming relaxation.     

一种基于匈牙利算法的二次分配问题求解方法

二次分配问题(Quadratic assignment problem,QAP)属于NP-hard组合优化难题.二次分配问题的线性化及下界计算方法,是求解二次分配问题的重要途径.以Frieze-Yadegar线性化模型和Gilmore-Lawler下界为基础,详细论述了二次分配问题线性化模型的结构特征,并分析了Gilmore-Lawler下界值往往远离目标函数最优值的原因.在此基础上,提出一种基于匈牙利算法的二次分配问题对偶上升下界求解法.通过求解QAPLIB中的部分实例,说明了方法的有效和可行性.     

一类线性乘积规划问题的分支定界缩减方法

提出了求解一类线性乘积规划问题的分支定界缩减方法, 并证明了算法的收敛性.在这个方法中, 利用两个变量乘积的凸包络技术, 给出了目标函数与约束函数中乘积的下界, 由此确定原问题的一个松弛凸规划, 从而找到原问题全局最优值的下界和可行解. 为了加快所提算法的收敛速度, 使用了超矩形的缩减策略. 数值结果表明所提出的算法是可行的.     

Multiple crane scheduling in a batch annealing process with no-delay constraints for machine unloading

In this work, we focus on the scheduling of multi-crane operations in an iron and steel enterprise for a two-stage batch annealing process. The first stage is the heating process, and the second stage is the cooling process. To start the heating (cooling) stage, a special machine called a furnace (cooler) must be loaded. The real constraints of no-delay machine unloading are defined as follows: once the heating (cooling) is completed, the furnace (cooler) must be unloaded by crane immediately. The goal is to schedule limited machines (furnaces and coolers) operated by multiple cranes to minimize the completion time of the last annealed coil (makespan). We formulate a mixed-integer linear programming model to address this problem. Certain feasible properties are identified to avoid crane conflicts and ensure that the machine unloading no-delay constraints are met. Based on these necessary conditions, we then present a heuristic algorithm with running time in connection with the number of cranes, coils and machines. A lower bound to the problem is also developed. Through theoretical analysis, we show the worst-case bound of our heuristic algorithm. The average performances of the solution approaches are computationally evaluated. The computational results show that the proposed heuristic algorithm is capable of generating good quality solutions.     

Equivalence of some LP-based lower bounds for the Golomb ruler problem

The Golomb Ruler problem consists in finding n integers such that all possible differences are distinct and such that the largest difference is minimum. We review three lower bounds based on linear programming that have been proposed in the literature for this problem, and propose a new one. We then show that these 4 lower bounds are equal. Finally we discuss some computational experience aiming at identifying the easiest lower bound to compute in practice.     

Tropical differential equations

Tropical differential equations are introduced and an algorithm is designed which tests solvability of a system of tropical linear differential equations within the complexity polynomial in the size of the system and in the absolute values of its coefficients. Moreover, we show that there exists a minimal solution, and the algorithm constructs it (in case of solvability). This extends a similar complexity bound established for tropical linear systems. In case of tropical linear differential systems in one variable a polynomial complexity algorithm for testing its solvability is designed.We prove also that the problem of solvability of a system of tropical non-linear differential equations in one variable is NP-hard, and this problem for arbitrary number of variables belongs to NP. Similar to tropical algebraic equations, a tropical differential equation expresses the (necessary) condition on the dominant term in the issue of solvability of a differential equation in power series.     

Computing the asymptotic worst-case of bin packing lower bounds

This paper addresses the issue of computing the asymptotic worst-case of lower bounds for the Bin Packing Problem. We introduce a general result that allows to bound the asymptotic worst-case performance of any lower bound for the problem and to derive for the first time the asymptotic worst-case of the well-known bound L₃ by Martello and Toth. We also show that the general result allows to easily derive the asymptotic worst-case of several lower bounds proposed in the literature.     

带非凸二次约束的二次比式和问题的全局优化算法(英文)

对带非凸二次约束的二次比式和问题(P)给出分枝定界算法,首先将问题(P)转化为其等价问题(Q),然后利用线性化技术,建立了(Q)松弛线性规划问题(RLP),通过对(RLP)可行域的细分及求解一系列线性规划问题,不断更新(Q)的上下界,从理论上证明了算法的收敛性,数值实验表明了算法的可行性和有效性.     

An exact algorithm to minimize mean squared deviation of job completion times about a common due date

We consider a deterministic n-job, single machine scheduling problem with the objective of minimizing the Mean Squared Deviation (MSD) of job completion times about a common due date (d). The MSD measure is non-regular and its value can decrease when one or more completion times increases. MSD problem is connected with the Completion Time Variance (CTV) problem and has been proved to be NP-hard. This problem finds application in situations where uniformity of service is important. We present an exact algorithm of pseudo-polynomial complexity, using ideas from branch and bound and dynamic programming. We propose a dominance rule and also develop a lower bound on MSD. The dominance rule and lower bound are effectively combined and used in the development of the proposed algorithm. The search space is explored using the breadth first branching strategy. The asymptotic space complexity of the algorithm is O(nd). Irrespective of the version of the problem – tightly constrained, constrained or unconstrained – the proposed algorithm provides optimal solutions for problem instances up to 1000 jobs size under different due date settings.     

Conic approximation to quadratic optimization with linear complementarity constraints

This paper proposes a conic approximation algorithm for solving quadratic optimization problems with linear complementarity constraints.We provide a conic reformulation and its dual for the original problem such that these three problems share the same optimal objective value. Moreover, we show that the conic reformulation problem is attainable when the original problem has a nonempty and bounded feasible domain. Since the conic reformulation is in general a hard problem, some conic relaxations are further considered. We offer a condition under which both the semidefinite relaxation and its dual problem become strictly feasible for finding a lower bound in polynomial time. For more general cases, by adaptively refining the outer approximation of the feasible set, we propose a conic approximation algorithm to identify an optimal solution or an \(\epsilon \)-optimal solution of the original problem. A convergence proof is given under simple assumptions. Some computational results are included to illustrate the effectiveness of the proposed algorithm.     

A branch and bound algorithm for globally solving a class of nonconvex programming problems

A branch and bound algorithm is proposed for globally solving a class of nonconvex programming problems (NP). For minimizing the problem, linear lower bounding functions (LLBFs) of objective function and constraint functions are constructed, then a relaxation linear programming is obtained which is solved by the simplex method and which provides the lower bound of the optimal value. The proposed algorithm is convergent to the global minimum through the successive refinement of linear relaxation of the feasible region and the solutions of a series of linear programming problems. And finally the numerical experiment is reported to show the feasibility and effectiveness of the proposed algorithm.     

线性比式和规划问题的输出空间分支定界算法

本文旨在针对线性比式和规划这一NP-Hard非线性规划问题提出新的全局优化算法.首先,通过引入p个辅助变量把原问题等价的转化为一个非线性规划问题,这个非线性规划问题的目标函数是乘积和的形式并给原问题增加了p个新的非线性约束,再通过构造凸凹包络的技巧对等价问题的目标函数和约束条件进行相应的线性放缩,构成等价问题的一个下界线性松弛规划问题,从而提出了一个求解原问题的分支定界算法,并证明了算法的收敛性.最后,通过数值结果比较表明所提出的算法是可行有效的.     

Resource augmented semi-online bounded space bin packing

We study on-line bounded space bin-packing in the resource augmentation model of competitive analysis. In this model, the on-line bounded space packing algorithm has to pack a list L of items with sizes in (0, 1], into a minimum number of bins of size b, b≥1. A bounded space algorithm has the property that it only has a constant number of active bins available to accept items at any point during processing. The performance of the algorithm is measured by comparing the produced packing with an optimal offline packing of the list L into bins of size 1. The competitive ratio then becomes a function of the on-line bin size b. Csirik and Woeginger studied this problem in [J. Csirik, G.J. Woeginger, Resource augmentation for online bounded space bin packing, Journal of Algorithms 44(2) (2002) 308-320] and proved that no on-line bounded space algorithm can perform better than a certain bound ρ(b) in the worst case. We relax the on-line condition by allowing a complete repacking within the active bins, and show that the same lower bound holds for this problem as well, and repacking may only allow one to obtain the exact best possible competitive ratio of ρ(b) having a constant number of active bins, instead of achieving this bound in the limit. We design a polynomial time on-line algorithm that uses three active bins and achieves the exact best possible competitive ratio ρ(b) for the given problem.     

Anticoloring and separation of graphs

An anticoloring of a graph is a coloring of some of the vertices, such that no two adjacent vertices are colored in distinct colors. The anticoloring problem seeks, roughly speaking, such colorings with many vertices colored in each color. We deal with the anticoloring problem for planar graphs and, using Lipton and Tarjan’s separation algorithm, provide an algorithm with some bound on the error. We also show that, to solve the anticoloring problem for general graphs, it suffices to solve it for connected graphs.     

Optimal allocation and processing time decisions on non-identical parallel CNC machines: ϵ-constraint approach

When the processing times of jobs are controllable, selected processing times affect both the manufacturing cost and the scheduling performance. A well known example for such a case that this paper specifically deals with is the turning operation on a CNC machine. Manufacturing cost of a turning operation is a nonlinear convex function of its processing time. In this paper, we deal with making optimal machine-job assignments and processing time decisions so as to minimize total manufacturing cost while the makespan being upper bounded by a known value, denoted as ?-constraint approach for a bicriteria problem. We then give optimality properties for the resulting single criterion problem. We provide alternative methods to compute cost lower bounds for partial schedules, which are used in developing an exact (branch and bound) algorithm. For the cases where the exact algorithm is not efficient in terms of computation time, we present a recovering beam search algorithm equipped with an improvement search procedure. In order to find improving search directions, the improvement search algorithm uses the proposed cost bounding properties. Computational results show that our lower bounding methods in branch and bound algorithm achieve a significant reduction in the search tree size that we need to traverse. Also, our recovering beam search and improvement search heuristics achieve solutions within 1% of the optimum on the average while they spent much less computational effort than the exact algorithm.     

A simplicial branch and duality bound algorithm for the sum of convex–convex ratios problem

This article presents a simplicial branch and duality bound algorithm for globally solving the sum of convex–convex ratios problem with nonconvex feasible region. To our knowledge, little progress has been made for globally solving this problem so far. The algorithm uses a branch and bound scheme where the Lagrange duality theory is used to obtain the lower bounds. As a result, the lower-bounding subproblems during the algorithm search are all ordinary linear programs that can be solved very efficiently. It has been proved that the algorithm possesses global convergence. Finally, the numerical experiments are given to show the feasibility of the proposed algorithm.     

A Branch and Bound Method for Solving Integer Separable Concave Problems

A branch and bound algorithm is proposed for solving integer separable concave problems. The method uses Lagrangian duality to obtain lower and upper bounds. We show that the dual program of a separable concave problem is a linear program. Moreover, we identify an excellent candidate to test on each region of the branch and we show an optimality sufficient condition for this candidate. Preliminary computational results are reported.