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.     

Using conical partition to globally maximizing the nonlinear sum of ratios

This article presents a global optimization algorithm for globally maximizing the sum of concave–convex ratios problem with a convex feasible region. The algorithm uses a branch and bound scheme where a concave envelope of the objective function is constructed to obtain an upper bound of the optimal value by using conical partition. As a result, the upper-bound subproblems during the algorithm search are all ordinary convex programs with less variables and constraints and do not grow in size from iterations to iterations in the computation procedure, and furthermore a new bounding tightening strategy is proposed such that the upper-bound convex relaxation subproblems are closer to the original nonconvex problem to enhance solution procedure. At last, some numerical examples are given to vindicate our conclusions.     

Using concave envelopes to globally solve the nonlinear sum of ratios problem

This article presents a branch and bound algorithm for globally solving the nonlinear sum of ratios problem (P). The algorithm works by globally solving a sum of ratios problem that is equivalent to problem (P). In the algorithm, upper bounds are computed by maximizing concave envelopes of a sum of ratios function over intersections of the feasible region of the equivalent problem with rectangular sets. The rectangular sets are systematically subdivided as the branch and bound search proceeds. Two versions of the algorithm, with convergence results, are presented. Computational advantages of these algorithms are indicated, and some computational results are given that were obtained by globally solving some sample problems with one of these algorithms.     

基于D.C.分解的一类箱型约束的非凸二次规划的新型分支定界算法

提出了一类求解带有箱约束的非凸二次规划的新型分支定界算法.首先,把原问题目标函数进行D.C.分解(分解为两个凸函数之差),利用次梯度方法,求出其线性下界逼近函数的一个最优值,也即原问题的一个下界.然后,利用全局椭球算法获得原问题的一个上界,并根据分支定界方法把原问题的求解转化为一系列子问题的求解.最后,理论上证明了算法的收敛性,数值算例表明算法是有效可行的.     

A flexible flowshop problem with total flow time minimization

In this study, we consider total flow time problem in a flexible flowshop environment. We develop a branch and bound algorithm to find the optimal schedule. The efficiency of the algorithm is enhanced by upper and lower bounds and a dominance criterion. Computational experience reveals that the algorithm solves moderate sized problems in reasonable solution times.     

A parallel branch and bound algorithm for solving large asymmetric traveling salesman problems

A parallel branch and bound algorithm that solves the asymmetric traveling salesman problem to optimality is described. The algorithm uses an assignment problem based lower bounding technique, subtour elimination branching rules, and a subtour patching algorithm as an upper bounding procedure. The algorithm is organized around a data flow framework for parallel branch and bound. The algorithm begins by converting the cost matrix to a sparser version in such a fashion as to retain the optimality of the final solution. Computational results are presented for three different classes of problem instances: (1) matrix elements drawn from a uniform distribution of integers for instances of size 250 to 10 000 cities, (2) instances of size 250 to 1000 cities that concentrate small elements in the upper left portion of the cost matrix, and (3) instances of size 300 to 3000 cities that are designed to confound neighborhood search heuristics.     

Job-shop local-search move evaluation without direct consideration of the criterion’s value

This article focuses on the evaluation of moves for the local search of the job-shop problem with the makespan criterion. We reason that the omnipresent ranking of moves according to their resulting value of a criterion function makes the local search unnecessarily myopic. Consequently, we introduce an alternative evaluation that relies on a surrogate quantity of the move’s potential, which is related to, but not strongly coupled with, the bare criterion. The approach is confirmed by empirical tests, where the proposed evaluator delivers a new upper bound on the well-known benchmark test yn2. The line of the argumentation also shows that by sacrificing accuracy the established makespan estimators unintentionally improve on the move evaluation in comparison to the exact makespan calculation, in contrast to the belief that the reliance on estimation degrades the optimization results.     

Uncertain constrained optimization by interval-oriented algorithm

This paper deals with an interval-oriented approach to solve general interval constrained optimization problems. Generally, this type of problems has infinitely many compromise solutions. The aim of this approach is to obtain one of such solutions with higher accuracy and lower computational cost. The proposed algorithm is nothing but a different kind of branch and bound algorithm with multi-section division criterion of the search region (or box). In the proposed technique, the prescribed/accepted region is divided into several distinct subregions and in each feasible subregion the interval objective function value is computed. Then the subregion containing the best objective value is found by applying a specific interval ranking rule defined with respect to the pessimistic decision makers’ point of view. The process is continued until the interval width for each variable in the accepted subregion is negligible. Finally, the algorithm converges to a compromise solution in interval form. To illustrate the method and also to test the efficiency as well as the effectiveness of the proposed algorithm, we have solved some numerical examples.     

A hybrid grouping genetic algorithm for bin packing

The grouping genetic algorithm (GGA) is a genetic algorithm heavily modified to suit the structure of grouping problems. Those are the problems where the aim is to find a good partition of a set or to group together the members of the set. The bin packing problem (BPP) is a well known NP-hard grouping problem: items of various sizes have to be grouped inside bins of fixed capacity. On the other hand, the reduction method of Martello and Toth, based on their dominance criterion, constitutes one of the best OR techniques for optimization of the BPP to date.In this article, we first describe the GGA paradigm as compared to the classic Holland-style GA and the ordering GA. We then show how the bin packing GGA can be enhanced with a local optimization inspired by the dominance criterion. An extensive experimental comparison shows that the combination yields an algorithm superior to either of its components.     

聚类分析在分枝定界法中的应用

针对界约束二次规划的分枝定界法中出现的紧、松弛策略,结合聚类分析方法,给出了新的剖分边的选取原则,把球约束二次规划作为子问题,使得原问题整体最优值的上、下界能较快的达到.     

101 Activities for Teaching Creativity and Problem Solving,Arthur B. VanGundy,Pfeiffer, J. Wiley,CA, USA (2004), p. xi + 391, £41.95, ISBN: 0-7879-7402-1

Many web sites (e.g. Hotmail, Yahoo) provide free services to the users while generating revenues from advertising. Advertising revenue is, therefore, critical for these sites. This in turn raises the question, how should advertisements at a web site be scheduled in a planning horizon to maximize revenue. Advertisements on the web are specified by geometry and display frequency and both of these factors need to be considered in developing a solution to the advertisement scheduling problem. Since this problem belongs to the class of NP-hard problems, we first develop a heuristic called LSMF to solve the problem. This heuristic is then combined with a genetic algorithm (GA) to develop a hybrid GA. The hybrid GA solution is first compared with the upper bound obtained by running CPLEX for the integer programming formulation of the problem. It is then also compared with an existing algorithm proposed in the literature. Our computational results show that the hybrid GA performs exceptionally well in the sense that it provides optimal or near optimal solution for a wide range of problem instances of realistic sizes and the improvements over existing algorithm are substantial. Finally we present a case study to illustrate how revenue could be significantly increased with a small improvement in the advertisement schedule. It is the first such study in this setup.     

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.     

Minimizing the sum of job completion times on capacitated two-parallel machines

This paper considers a scheduling problem with two identical parallel machines. One has unlimited capacity; the other can only run for a fixed time. A given set of jobs must be scheduled on the two machines with the goal of minimizing the sum of their completion times. The paper proposes an optimal branch and bound algorithm which employs three powerful elements, including an algorithm for computing the upper bound, a lower bound algorithm, and a fathoming condition. The branch and bound algorithm was tested on problems of various sizes and parameters. The results show that the algorithm is quite efficient to solve all the test problems. In particular, the total computation time for the hardest problem is less than 0.1 second for a set of 100 problem instances. An important finding of the tests is that the upper bound algorithm can actually find optimal solutions to a quite large number of problems.     

A genetic algorithm for the two-stage supply chain distribution problem associated with a fixed charge

This paper considers a two-stage distribution problem of a supply chain that is associated with a fixed charge. Two kinds of cost are involved in this problem: a continuous cost that linearly increases with the amount transported between a source and a destination, and secondly, a fixed charge, that incurs whenever there exists a transportation of a non-zero quantity between a source and a destination. The objective criterion is the minimisation of the total cost of distribution. A genetic algorithm (GA) that belongs to evolutionary search heuristics is proposed and illustrated. The proposed methodology is evaluated for its solution quality by comparing it with the approximate and lower bound solutions. Thus, the comparison reveals that the GA generates better solution than the approximation method and is capable of providing solution either equal or closer to the lower bound solution of the problem.     

Strong planning and forecast horizons for a model with simultaneous price and production decisions

In this paper, we have solved a general inventory model with simultaneous price and production decisions. Both linear and non-linear (strictly convex) production cost cases are treated. Upper and lower bounds are imposed on state as well as control variables. The problem is solved by using the Lagrangian form of the maximum principle. Strong planning and strong forecast horizons are obtained. These arise when the state variable reaches its upper or lower bound. The existence of these horizons permits the decomposition of the whole problem into a set of smaller problems, which can be solved separately, and their solutions put together to form a complete solution to the problem. Finally, we derive a forward branch and bound algorithm to solve the problem. The algorithm is illustrated with a simple example.     

The efficiency of hybrid mutation genetic algorithm for the travelling salesman problem

In this paper, we present an efficient genetic algorithm (GA) for solving the travelling salesman problem (TSP) as a combinatorial optimization problem. In our computational model, we propose a complete subtour exchange crossover that does not break as some good subtours as possible, because the good subtours are worth preserving for descendants. Generally speaking, global search GA is considered to be better approaches than local searches. However, it is necessary to strengthen the ability of local search as well as global ones in order to increase a GA total efficiency. In this study, our GA applies a stochastic hill climbing procedure in the mutation process of the GA. Experimental results showed that the GA leads good convergence as high as 99 percent even for 500 cities TSP.     

A mathematical model of market competition

We consider a mathematical model of decision making by a company attempting to win a market share. We assume that the company releases its products to the market under the competitive conditions that another company is making similar products. Both companies can vary the kinds of their products on the market as well as the prices in accordance with consumer preferences. Each company aims to maximize its profit. A mathematical statement of the decision-making problem for the market players is a bilevel mathematical programming problem that reduces to a competitive facility location problem. As regards the latter, we propose a method for finding an upper bound for the optimal value of the objective function and an algorithm for constructing an approximate solution. The algorithm amounts to local ascent search in a neighborhood of a particular form, which starts with an initial approximate solution obtained simultaneously with an upper bound. We give a computational example of the problem under study which demonstrates the output of the algorithm.     

A finite concave minimization algorithm using branch and bound and neighbor generation

In this article we present a new finite algorithm for globally minimizing a concave function over a compact polyhedron. The algorithm combines a branch and bound search with a new process called neighbor generation. It is guaranteed to find an exact, extreme point optimal solution, does not require the objective function to be separable or even analytically defined, requires no nonlinear computations, and requires no determinations of convex envelopes or underestimating functions. Linear programs are solved in the branch and bound search which do not grow in size and differ from one another in only one column of data. Some preliminary computational experience is also presented.     

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.     

A Branch and Contract Algorithm for Problems with Concave Univariate, Bilinear and Linear Fractional Terms

A new deterministic branch and bound algorithm is presented in this paper for the global optimization of continuous problems that involve concave univariate, bilinear and linear fractional terms. The proposed algorithm, the branch and contract algorithm, relies on the use of a bounds-contraction subproblem that aims at reducing the size of the search region by eliminating portions of the domain in which the objective function takes only values above a known upper bound. The solution of contraction subproblems at selected branch and bound nodes is performed within a finite contraction operation that helps reducing the total number of nodes in the branch and bound solution tree. The use of the proposed algorithm is illustrated with several numerical examples.