Comments on “A bicriteria flowshop scheduling problem with setup times”

The aim of this paper is to point out that the integer programming model proposed by Eren and Güner [T. Eren, E. Güner, A bicriteria flowshop scheduling problem with setup times, Appl. Math. Comput. 183 (2006) 1292-1300] is incorrect. We propose a new integer programming model for the same scheduling problem based on their model.     

Notes on “A bicriteria flowshop scheduling with a learning effect”

The aim of this paper is to point out that the integer programming model proposed by Eren and Güner [T. Eren, E. Güner, A bicriteria flowshop scheduling with a learning effect, Applied Mathematical Modelling 32 (2008) 1719–1733] is incorrect. We propose a mixed 0–1 programming model for the same scheduling problem based on their model.     

Comments on “A bicriteria m-machine flowshop scheduling with sequence-dependent setup times”

The purpose of this paper is to point out that the programming model proposed by Eren [A bicriteria m-machine flowshop scheduling with sequence-dependent setup times, Appl. Math. Model. 34 (2010) 284–293] is incorrect. We propose a mixed 0–1 programming model for the same scheduling problem based on their model.     

A new method of solving nonlinear mathematical programming problems involving r-invex functions

A new approach to a solution of a nonlinear constrained mathematical programming problem involving r-invex functions with respect to the same function η is introduced. An η-approximated problem associated with an original nonlinear mathematical programming problem is presented that involves η-approximated functions constituting the original problem. The equivalence between optima points for the original mathematical programming problem and its η-approximated optimization problem is established under r-invexity assumption.     

Parametric Duality Models for Semi-infinite Discrete Minmax Fractional Programming Problems Involving Generalized （η,ρ）-Invex Functions

A semi-infinite programming problem is a mathematical programming problem with a finite number of variables and infinitely many constraints. Duality theories and generalized convexity concepts are important research topics in mathematical programming. In this paper, we discuss a fairly large number of paramet- ric duality results under various generalized （η,ρ）-invexity assumptions for a semi-infinite minmax fractional programming problem.     

A mathematical programming approach to clusterwise regression model and its extensions

The clusterwise regression model is used to perform cluster analysis within a regression framework. While the traditional regression model assumes the regression coefficient (β) to be identical for all subjects in the sample, the clusterwise regression model allows β to vary with subjects of different clusters. Since the cluster membership is unknown, the estimation of the clusterwise regression is a tough combinatorial optimization problem. In this research, we propose a “Generalized Clusterwise Regression Model” which is formulated as a mathematical programming (MP) problem. A nonlinear programming procedure (with linear constraints) is proposed to solve the combinatorial problem and to estimate the cluster membership and β simultaneously. Moreover, by integrating the cluster analysis with the discriminant analysis, a clusterwise discriminant model is developed to incorporate parameter heterogeneity into the traditional discriminant analysis. The cluster membership and discriminant parameters are estimated simultaneously by another nonlinear programming model.     

An η-approximation approach to duality in mathematical programming problems involving r-invex functions

An η-approximation approach introduced by Antczak [T. Antczak, A new method of solving nonlinear mathematical programming problems involving r-invex functions, J. Math. Anal. Appl. 311 (2005) 313-323] is used to obtain a solution Mond-Weir dual problems involving r-invex functions. η-Approximated Mond-Weir dual problems are introduced for the η-approximated optimization problem constructed in this method associated with the original nonlinear mathematical programming problem. By the help of η-approximated dual problems various duality results are established for the original mathematical programming problem and its original Mond-Weir duals.     

A note on duality gaps in linear programming over convex sets

For certain types of mathematical programming problems, a related dual problem can be constructed in which the objective value of the dual problem is equal to the objective function of the given problem. If these two problems do not have equal values, a duality gap is said to exist. No such gap exists for pairs of ordinary dual linear programming problems, but this is not the case for linear programming problems in which the nonnegativity conditionx ? 0 is replaced by the condition thatx lies in a certain convex setK. Duffin (Ref. 1) has shown that, whenK is a cone and a certain interiority condition is fulfilled, there will be no duality gap. In this note, we show that no duality gap exists when the interiority condition is satisfied andK is an arbitrary closed convex set inR ⁿ .     

Comment on “Complete solution to a conjecture on Randić index”

In this paper we will show that the proof of Theorem 2.1 from “Complete solution to a conjecture on Randi? index”, by Xueliang Li, Bolian Liu and Jianxi Liu, European Journal of Operational Research 200, Issue 1, (2010), 9–13, is not correct. They tried to prove the conjecture given by M. Aouchiche, P. Hansen in “On a conjecture about the Randi? index” (Discrete Mathematics, 307 (2), 2007, 262–265), but they failed in it. The mathematical model given by them is a problem of quadratic programming which they tried to solve by wrong use of linear programming. This error invalidates all further work.     

Minimising total average cycle stock subject to practical constraints

Silver and Moon (J Opl Res Soc 50(8) (1999) 789–796) address the problem of minimising total average cycle stock subject to two practical constraints. They provide a dynamic programming formulation for obtaining an optimal solution and propose a simple and efficient heuristic algorithm. Hsieh (J Opl Res Soc 52(4) (2001) 463–470) proposes a 0–1 linear programming approach to the problem and a simple heuristic based on the relaxed 0–1 programming formulation. We show in this paper that the formulation of Hsieh can be improved for solving very large size instances of this inventory problem. So the mathematical approach is interesting for several reasons: the definition of the model is simple, its implementation is immediate by using a mathematical programming language together with a mixed integer programming software and the performance of the approach is excellent. Computational experiments carried out on the set of realistic examples considered in the above references are reported. We also show that the general framework for modelling given by mixed integer programming allows the initial model to be extended in several interesting directions.     

An extended assignment problem considering multiple inputs and outputs

The existing assignment problems for assigning n jobs to n individuals are limited to the considerations of cost or profit incurred by each possible assignment. However, in real applications, various inputs and outputs are usually concerned in an assignment problem, such as a general decision-making problem. This paper develops a procedure for resolving assignment problems with multiple incommensurate inputs and outputs for each possible assignment. The concept of the relative efficiency in using various resources, instead of cost or profit, is adopted for each possible assignment of the problem. Data envelopment analysis (DEA) is employed in this paper to measure the efficiency of one assignment relative to that of the others according to a set of decision-making units. A composite efficiency index, consisting of two kinds of relative efficiencies under different comparison bases, is defined to serve as the performance measurement of each possible assignment in the problem formulation. A mathematical programming model for the extended assignment problem is proposed, which is then expressed as a classical integer linear programming model to determine the assignments with the maximum efficiency. A numerical example is used to demonstrate the approach.     

A new exact exponential penalty function method and nonconvex mathematical programming

A new exact penalty function method, called the l₁ exact exponential penalty function method, is introduced. In this approach, the so-called the exponential penalized optimization problem with the l₁ exact exponential penalty function is associated with the original optimization problem with both inequality and equality constraints. The l₁ exact exponential penalty function method is used to solve nonconvex mathematical programming problems with r-invex functions (with respect to the same function η). The equivalence between sets of optimal solutions of the original mathematical programming problem and of its associated exponential penalized optimization problem is established under suitable r-invexity assumption. Also lower bounds on the penalty parameter are given, for which above these values, this result is true.     

A branch&cut algorithm for the maximum common edge subgraph problem

We investigate the Maximum Common Edge Subgraph Problem (MCES) defined as follows. Given two graphs G and H with the same number of vertices, find a common subgraph of G and H (not necessary induced) with the maximum number of edges. This problem arises in parallel programming environments, and was first defined by Bokhari in [S. Bokhari, On the mapping problem, IEEE Trans. Comput., C-30(3), 1981]. We present a new integer programming formulation for the MCES problem and carry out a polyhedral investigation of this model. A number of valid inequalities are identified, most of which are facet-defining. We also report on computational experiments.     

Sonar array element location: A hybrid expert systems application

The design of many engineering systems, including sonar systems, has evolved (thanks in part to methods of operational research such as mathematical programming and simulation) from an almost strictly art form into a relatively scientific endeavor. Today we see, more and more, the use of, in particular, mathematical programming methods for the development of ‘optimally’ designed engineering hardware. Unfortunately, such designs are ‘optimal’ only under the unrealistic assumption of an error free model and the ability to develop a corresponding perfect design. That is, they fail to take into regard the often severe impact of the limitations of real world design tolerances. And this failure is becoming increasing more important as we begin to address modern day systems with their substantially increased size, sophistication and sensitivity.In this paper, we discuss the problem, with specific focus on the task of the development of sonar arrays. We present a hybrid methodology, combining expert systems with heuristic programming, that permits the engineer to incorporate the results of the mathematical programming solution while still using real world, imperfect components. And we present results as obtained on actual hardware.     

Examining the effect of second-order terms in mathematical programming approaches to the classification problem

Research on mathematical programming approaches to the classification problem has focused almost exclusively on linear discriminant functions with only first-order terms. While many of these first-order models have displayed excellent classificatory performance when compared to Fisher's linear discriminant method, they cannot compete with Smith's quadratic discriminant method on certain data sets. In this paper, we investigate the appropriateness of including second-order terms in mathematical programming models. Various issues are addressed, such as performance of models with small to moderate sample size, need for crossproduct terms, and loss of power by the mathematical programming models under conditions ideal for the parametric procedures. A simulation study is conducted to assess the relative performance of first-order and second-order mathematical programming models to the parametric procedures. The simulation study indicates that mathematical programming models using polynomial functions may be prone to overfitting on the training samples which in turn may cause rather poor fits on the validation samples. The simulation study also indicates that inclusion of cross-product terms may hurt a polynomial model's accuracy on the validation samples, although omission of them means that the model is not invariant to nonsingular transformations of the data.     

A Probabilistic Objective Function for R & D Portfolio Selection

This paper contributes to the literature on the problem of allocation of resources to a set of risky investments. Our objective is to develop the ideas in the context of a research and development laboratory. A mathematical programming approach to the resource allocation problem is taken, and various forms for an objective function under risk are discussed. A probabilistic objective function appropriate to R & D is isolated and tested on a small hypothetical example. Parametric linear programming is used to yield a near-optimum allocation.     

Probabilistic Networks and R & D Portfolio Selection

Mathematical programming methods have been suggested and used as an aid to R & D project portfolio selection. One of the main criticisms of the use of such models is that the stochastic nature of the problem has been largely ignored. This paper presents a method which takes into account the stochastic nature of resource requirements and project benefits, using a combination of probabilistic networks, simulation and mathematical programming. A case study based on data from an industrial R & D laboratory is presented and compared with the use of expected value methods. The results of the study indicate that in this particular case the deterministic linear programming solution is robust.     

A bicriteria solid transportation problem with fixed charge under stochastic environment

In this paper, a bicriteria solid transportation problem with stochastic parameters is investigated. Three mathematical models are constructed for the problem, including expected value goal programming model, chance-constrained goal programming model and dependent-chance goal programming model. A hybrid algorithm is also designed based on the random simulation algorithm and tabu search algorithm to solve the models. At last, some numerical experiments are presented to show the performance of models and algorithm.     

On an (<Emphasis Type="Italic">s,Q</Emphasis>) Production Policy for an Integrated Inventory Model for a Single Product with Linearly Increasing Demand

We analyse an (s, Q) production policy for an inventory system consisting of a single finished product and the raw materials used for manufacturing it, and where the demand rate of the product increases linearly with time. We formulate a mathematical programming model with the objective of minimizing total inventory cost per unit time. The problem of grouping raw materials optimally so that common replenishment periods may be used is considered. Solution procedures are developed, and numerical examples are presented.     

A mathematical programming approach to the computation of the omega invariant of a numerical semigroup

In this paper we present a mathematical programming formulation for the ω-invariant of a numerical semigroup for each of its minimal generators which is an useful index in commutative algebra (in particular in factorization theory) to analyze the primality of the elements in the semigroup. The model consists of solving a problem of optimizing a linear function over the efficient set of a multiobjective linear integer program. We offer a methodology to solve this problem and we provide some computational experiments to show the efficiency of the proposed algorithm.