An achievement rate approach to linear programming problems with an interval objective function

In this paper, we focus on a treatment of a linear programming problem with an interval objective function. From the viewpoint of the achievement rate, a new solution concept, the maximin achievement rate solution, is proposed. Nice properties of this solution are shown: a maximin achievement rate solution is necessarily optimal when a necessarily optimal solution exists, and if not, then it is still a possibly optimal solution. An algorithm for a maximin achievement rate solution is proposed based on a relaxation procedure together with a simplex method. A numerical example is given to demonstrate the proposed solution algorithm.     

Location with acceleration–deceleration distance

In this paper we investigate a model where travel time is not necessarily proportional to the distance. Every trip starts at speed zero, then the vehicle accelerates to a cruising speed, stays at the cruising speed for a portion of the trip and then decelerates back to a speed of zero. We define a time equivalent distance which is equal to the travel time multiplied by the cruising speed. This time equivalent distance is referred to as the acceleration–deceleration (A–D) distance. We prove that every demand point is a local minimum for the Weber problem defined by travel time rather than distance. We propose a heuristic approach employing the generalized Weiszfeld algorithm and an optimal approach applying the Big Triangle Small Triangle global optimization method. These two approaches are very efficient and problems of 10,000 demand points are solved in about 0.015 seconds by the generalized Weiszfeld algorithm and in about 1 minute by the BTST technique. When the generalized Weiszfeld algorithm was repeated 1000 times, the optimal solution was found at least once for all test problems.     

Optimal berth allocation and time-invariant quay crane assignment in container terminals

Due to the dramatic increase in the world’s container traffic, the efficient management of operations in seaport container terminals has become a crucial issue. In this work, we focus on the integrated planning of the following problems faced at container terminals: berth allocation, quay crane assignment (number), and quay crane assignment (specific). First, we formulate a new binary integer linear program for the integrated solution of the berth allocation and quay crane assignment (number) problems called BACAP. Then we extend it by incorporating the quay crane assignment (specific) problem as well, which is named BACASP. Computational experiments performed on problem instances of various sizes indicate that the model for BACAP is very efficient and even large instances up to 60 vessels can be solved to optimality. Unfortunately, this is not the case for BACASP. Therefore, to be able to solve large instances, we present a necessary and sufficient condition for generating an optimal solution of BACASP from an optimal solution of BACAP using a post-processing algorithm. In case this condition is not satisfied, we make use of a cutting plane algorithm which solves BACAP repeatedly by adding cuts generated from the optimal solutions until the aforementioned condition holds. This method proves to be viable and enables us to solve large BACASP instances as well. To the best of our knowledge, these are the largest instances that can be solved to optimality for this difficult problem, which makes our work applicable to realistic problems.     

Two new algorithms for solving optimization problems with one linear objective function and finitely many constraints of fuzzy relation inequalities

This paper studies the optimization model of a linear objective function subject to a system of fuzzy relation inequalities (FRI) with the max-Einstein composition operator. If its feasible domain is non-empty, then we show that its feasible solution set is completely determined by a maximum solution and a finite number of minimal solutions. Also, an efficient algorithm is proposed to solve the model based on the structure of FRI path, the concept of partial solution, and the branch-and-bound approach. The algorithm finds an optimal solution of the model without explicitly generating all the minimal solutions. Some sufficient conditions are given that under them, some of the optimal components of the model are directly determined. Some procedures are presented to reduce the search domain of an optimal solution of the original problem based on the conditions. Then the reduced domain is decomposed (if possible) into several sub-domains with smaller dimensions that finding the components of the optimal solution in each sub-domain is very easy. In order to obtain an optimal solution of the original problem, we propose another more efficient algorithm which combines the first algorithm, these procedures, and the decomposition method. Furthermore, sufficient conditions are suggested that under them, the problem has a unique optimal solution. Also, a comparison between the recently proposed algorithm and the known ones will be made.     

A mathematical programming-based scheduling framework for multitasking environments

A new paradigm along with a mixed (binary) integer-linear programming model is developed for scheduling tasks in multitasking environments, for which the number of completed tasks is not a good measure. One special case falls into the realm of deteriorating jobs. Polynomial time optimal solution algorithms are presented for this and one other special case. As the complexity of the original problem is believed to be strongly NP-hard, an efficient solution algorithm, based on tabu search, is developed to solve the problem. Small, medium, and large size problems are solved, and the solution obtained from the algorithm is compared with that of the optimal solution or the upper bound found from using the Lagrangian relaxation. Where it was measurable, the search algorithm gave quantifiably good quality solutions, and in all cases it had a much better time efficiency than the branch-and-bound enumeration method. A detailed statistical experiment, based on the split-plot design, is developed to identify the characteristics of the tabu search algorithm, thus guaranteeing a solution that is significantly better in quality. A conjecturing technique is introduced for problems with very large planning horizons. This technique had remarkable time efficiency with no apparent loss of quality.     

A modified method for constructing efficient solutions structure of MOLP

This paper deals with a recently proposed algorithm for obtaining all weak efficient and efficient solutions in a multi objective linear programming (MOLP) problem. The algorithm is based on solving some weighted sum problems, and presents an easy and clear solution structure. We first present an example to show that the algorithm may fail when at least one of these weighted sum problems has not a finite optimal solution. Then, the algorithm is modified to overcome this problem. The modified algorithm determines whether an efficient solution exists for a given MOLP and generates the solution set correctly (if exists) without any change in the complexity.     

Column Generation Algorithms for Nonlinear Optimization,I: Convergence Analysis

Column generation is an increasingly popular basic tool for the solution of large-scale mathematical programming problems. As problems being solved grow bigger, column generation may however become less efficient in its present form, where columns typically are not optimizing, and finding an optimal solution instead entails finding an optimal convex combination of a huge number of them. We present a class of column generation algorithms in which the columns defining the restricted master problem may be chosen to be optimizing in the limit, thereby reducing the total number of columns needed. This first article is devoted to the convergence properties of the algorithm class, and includes global (asymptotic) convergence results for differentiable minimization, finite convergence results with respect to the optimal face and the optimal solution, and extensions of these results to variational inequality problems. An illustration of its possibilities is made on a nonlinear network flow model, contrasting its convergence characteristics to that of the restricted simplicial decomposition (RSD) algorithm.     

Global optimization for sum of geometric fractional functions

This paper presents an efficient branch and bound algorithm for globally solving sum of geometric fractional functions under geometric constraints, which arise in various practical problems. By using an equivalent transformation and a new linear relaxation technique, a linear relaxation programming problem of the equivalent problem is obtained. The proposed algorithm is convergent to the global optimal solution by means of the subsequent solutions of a series of linear programming problems. Numerical results are reported to show the feasibility of our algorithm.     

Minimax regret solution to multiobjective linear programming problems with interval objective functions coefficients

The current paper focuses on a multiobjective linear programming problem with interval objective functions coefficients. Taking into account the minimax regret criterion, an attempt is being made to propose a new solution i.e. minimax regret solution. With respect to its properties, a minimax regret solution is necessarily ideal when a necessarily ideal solution exists; otherwise it is still considered a possibly weak efficient solution. In order to obtain a minimax regret solution, an algorithm based on a relaxation procedure is suggested. A numerical example demonstrates the validity and strengths of the proposed algorithm. Finally, two special cases are investigated: the minimax regret solution for fixed objective functions coefficients as well as the minimax regret solution with a reference point. Some of the characteristic features of both cases are highlighted thereafter.     

Cutting and packing problems for irregular objects with continuous rotations: mathematical modelling and non-linear optimization

We further improve our methodology for solving irregular packing and cutting problems. We deal with an accurate representation of objects bounded by circular arcs and line segments and allow their continuous rotations and translations within rectangular and circular containers. We formulate a basic irregular placement problem which covers a wide spectrum of packing and cutting problems. We provide an exact non-linear programming (NLP) model of the problem, employing ready-to-use phi-functions. We develop an efficient solution algorithm to search for local optimal solutions for the problem in a reasonable time. The algorithm reduces our problem to a sequence of NLP subproblems and employs optimization procedures to generate starting feasible points and feasible subregions. Our algorithm allows us to considerably reduce the number of inequalities in NLP subproblems. To show the benefits of our methodology we give computational results for a number of new challenger and the best known benchmark instances.     

An efficient genetic algorithm for the traveling salesman problem with precedence constraints

The traveling salesman problem with precedence constraints (TSPPC) is one of the most difficult combinatorial optimization problems. In this paper, an efficient genetic algorithm (GA) to solve the TSPPC is presented. The key concept of the proposed GA is a topological sort (TS), which is defined as an ordering of vertices in a directed graph. Also, a new crossover operation is developed for the proposed GA. The results of numerical experiments show that the proposed GA produces an optimal solution and shows superior performance compared to the traditional algorithms.     

A nonisolated optimal solution for special reverse convex programming problems

In this paper, an efficient algorithm is proposed for globally solving special reverse convex programming problems with more than one reverse convex constraints. The proposed algorithm provides a nonisolated global optimal solution which is also stable under small perturbations of the constraints, and it turns out that such an optimal solution is adequately guaranteed to be feasible and to be close to the actual optimal solution. Convergence of the algorithm is shown and the numerical experiment is given to illustrate the feasibility of the presented algorithm.     

模糊多目标半定规划

This paper first applies the fuzzy set theory to multi-objective semi-definite program-ming （MSDP）, and proposes the fuzzy multi-objective semi-definite programming （FMSDP） model whose optimal efficient solution is defined for the first time, too. By constructing a membership function, the FMSDP is translated to the MSDP. Then we prove that the optimal efficient solution of FMSDP is consistent with the efficient solution of MSDP and present the optimality condition about these programming. At last, we give an algorithm for FMSDP by introducing a new membership function and a series of transformation.     

Strongly sub-feasible direction method for constrained optimization problems with nonsmooth objective functions

In this paper, we propose a strongly sub-feasible direction method for the solution of inequality constrained optimization problems whose objective functions are not necessarily differentiable. The algorithm combines the subgradient aggregation technique with the ideas of generalized cutting plane method and of strongly sub-feasible direction method, and as results a new search direction finding subproblem and a new line search strategy are presented. The algorithm can not only accept infeasible starting points but also preserve the “strong sub-feasibility” of the current iteration without unduly increasing the objective value. Moreover, once a feasible iterate occurs, it becomes automatically a feasible descent algorithm. Global convergence is proved, and some preliminary numerical results show that the proposed algorithm is efficient.     

Simplicial Branch-and-Reduce Algorithm for Convex Programs with a Multiplicative Constraint

Optimization problems that involve products of convex functions in the objective function or in the constraints arise in a variety of applications. These problems are difficult global optimization problems. During the past 15 years, however, a number of practical algorithms have been proposed for globally solving these types of problems. In this article, we present and validate a branch-and-reduce algorithm for finding a global optimal solution to a convex program that contains an additional constraint on the product of several convex functions. To globally solve this problem, the algorithm instead globally solves an equivalent master problem. At any stage of the algorithm, a disconnected set consisting of a union of simplices is constructed. This set is guaranteed to contain a portion of the boundary of the feasible region of the master problem where a global optimal solution lies. The algorithm uses a new branch-and-reduce scheme to iteratively reduce the sizes of these sets until a global optimal solution is found. Several potential computational advantages of the algorithm are explained, and a numerical example is solved.     

An Algorithm for Solving Optimization Problems with One Linear Objective Function and Finitely Many Constraints of Fuzzy Relation Inequalities

An optimization model with one linear objective function and fuzzy relation equation constraints was presented by Fang and Li (1999) as well as an efficient solution procedure was designed by them for solving such a problem. A more general case of the problem, an optimization model with one linear objective function and finitely many constraints of fuzzy relation inequalities, is investigated in this paper. A new approach for solving this problem is proposed based on a necessary condition of optimality given in the paper. Compared with the known methods, the proposed algorithm shrinks the searching region and hence obtains an optimal solution fast. For some special cases, the proposed algorithm reaches an optimal solution very fast since there is only one minimum solution in the shrunk searching region. At the end of the paper, two numerical examples are given to illustrate this difference between the proposed algorithm and the known ones.     

A revisit of numerical approach for solving linear fractional programming problem in a fuzzy environment

In the article, Veeramani and Sumathi [10] presented an interesting algorithm to solve a fully fuzzy linear fractional programming (FFLFP) problem with all parameters as well as decision variables as triangular fuzzy numbers. They transformed the FFLFP problem under consideration into a bi-objective linear programming (LP) problem, which is then converted into two crisp LP problems. In this paper, we show that they have used an inappropriate property for obtaining non-negative fuzzy optimal solution of the same problem which may lead to the erroneous results. Using a numerical example, we show that the optimal fuzzy solution derived from the existing model may not be non-negative. To overcome this shortcoming, a new constraint is added to the existing fuzzy model that ensures the fuzzy optimal solution of the same problem is a non-negative fuzzy number. Finally, the modified solution approach is extended for solving FFLFP problems with trapezoidal fuzzy parameters and illustrated with the help of a numerical example.     

Scheduling preemptive open shops to minimize total tardiness

This article considers the problem of scheduling preemptive open shops to minimize total tardiness. The problem is known to be NP-hard. An efficient constructive heuristic is developed for solving large-sized problems. A branch-and-bound algorithm that incorporates a lower bound scheme based on the solution of an assignment problem as well as various dominance rules are presented for solving medium-sized problems. Computational results for the 2-machine case are reported. The branch-and-bound algorithm can handle problems of up to 30 jobs in size within a reasonable amount of time. The solution obtained by the heuristic has an average deviation of less than 2% from the optimal value, while the initial lower bound has an average deviation of less than 11% from the optimal value. Moreover, the heuristic finds approved optimal solutions for over 65% of the problems actually solved.     

An Objective Penalty Function of Bilevel Programming

Penalty methods are very efficient in finding an optimal solution to constrained optimization problems. In this paper, we present an objective penalty function with two penalty parameters for inequality constrained bilevel programming under the convexity assumption to the lower level problem. Under some conditions, an optimal solution to a bilevel programming defined by the objective penalty function is proved to be an optimal solution to the original bilevel programming. Moreover, based on the objective penalty function, an algorithm is developed to obtain an optimal solution to the original bilevel programming, with its convergence proved under some conditions.     

基于灰色理论的价格控制问题及其算法

针对排污收费的最优定价问题,提出了基于灰色理论的价格控制问题,并给出了该问题的模型及相关的定理。在约束域为非空紧集的条件下,证明了漂移型价格控制问题的最优解一定可以在约束域的极点达到。针对漂移型价格控制问题,采用价格控制问题的搜索算法的求解技术,把灰参数看做一个新的决策变量,将该问题转化为多个含参数的非线性规划问题。最后,通过一算例验证了模型及求解方法的有效性。