A mathematical programming viewpoint for solving the ultimate pit problem

In 1965 Helmut Lerchs and Ingo Grossmann presented to the mining community an algorithm to find the optimum design for an open pit mine. In their words, “the objective is to design the contour of a pit so as to maximize the difference between total mine value of the ore extracted and the total extraction cost of ore and waste”. They modeled the problem in graph theoretic terms and showed that an optimal solution of the ultimate pit problem is equivalent to finding the maximum closure of their graph based model. In this paper, we develop a network flow algorithm based on the dual to solve the same problem. We show how this algorithm is closely related to Lerchs and Grossmann's and how the steps in their algorithm can be viewed in mathematical programming terms. This analysis adds insight to the algorithm of Lerchs and Grossmann and shows where it can be made more efficient. As in the case Lerchs and Grossmann, our algorithm allows us to use very efficient data structures based on graphs that represent the data and constraints.. 1782 1528 V 3     

General mathematical programming formulations for the statistical classification problem

The study presents mathematical programming formulations for the statistical classification problem. The formulations maximize the number of observations that are properly categorized. The models are easily modified for any number of categories of classification. The formulation were tested on a standard problem.     

A mathematical programming problem with singular mixed pointwise-integral constraints

We investigate an infinite dimensional optimization problem which constraints are singular integral-pointwise ones. We give some partial results of existence for a solution in some particular cases. However, the lack of compactness, even in L¹ prevents to conclude in the general case. We give an existence result for a weak solution (as a measure) that we are able to describe. The regularity of such a solution is still an open problem.     

A mathematical model for a problem in botany

In this paper, an attempt is made to construct a mathematical model to present a picture of a biological system, namely the evolutionary status of secondary phloem in dicotyledonous plants. In this study 16 parameters have been taken into account. The biological nature of each parameter leads to a mathematical relation and this relationship is exploited to compute a number (conveniently made to lie between 0 and 100) which represents the advance feature of the plant. Thus, given two plants A and B, these computations show whether A is more advanced than B or not. In this communication a set of 18 plants are considered.     

A mathematical programming model for optimal service selection in the airline industry

Airlines, offering tour plans, have to select a few gateways (airports) to which the air tickets for tour plans are issued. A number of alternate gateways are available for different tour plans, and the selection of gateways is influenced by the perceived preferences of tourists for these and competing plans. In this paper, using the ordinal preferences of tourists, we formulate the problem of gateway selection as an integer program. The solution procedure presented for this integer program is computationally attractive as we exploit the special structure of our formulation. The model proposed is general, and can be applied to the simultaneous positioning of several new products or services.     

Generalized mathematical programming problem in system design

Translated from Sistemnoe Programmirovanie i Modeli Issledovaniya Operatsii, pp. 183–194, 1993.     

A remark on a regularity assumption for the mathematical programming problem in Banach spaces

This paper deals with a recently proposed Slater-like regularity condition for the mathematical programming problem in infinite-dimensional vector spaces (Ref. 1). The attractive feature of this constraint qualification is the fact that it can be considered as a condition only on theactive part of the constraint. We prove that the studied regularity condition is equivalent to the regularity assumption normally used in the study of the mathematical programming problem in infinite-dimensional vector spaces.     

A finite procedure to generate feasible points for the extreme point mathematical programming problem

We present a finite cutting plane procedure to generate feasible points for the extreme point mathematical programming (EPMP) problem, and show that this procedure can be used for solving nonconvex programs of other type as well. A finite method is described for the EPMP problem. Computational experience of EPMP and concave minization problems is also provided.     

Hybrid large neighborhood search for the bus rapid transit route design problem

Due to an increasing demand for public transportation and intra-urban mobility, an efficient organization of public transportation has gained significant importance in the last decades. In this paper we present a model formulation for the bus rapid transit route design problem, given a fixed number of routes to be offered. The problem can be tackled using a decomposition strategy, where route design and the determination of frequencies and passenger flows will be dealt with separately. We propose a hybrid metaheuristic based on a combination of Large Neighborhood Search (LNS) and Linear Programming (LP). The algorithm as such is iterative. Decision upon the design of routes will be handled using LNS. The resulting passenger flows and frequencies will be determined by solving a LP. The solution obtained may then be used to guide the exploration of new route designs in the following iterations within LNS. Several problem specific operators are suggested and have been tested. The proposed algorithm compares extremely favorable and is able to obtain high quality solutions within short computational times.     

Regularity and stability for the mathematical programming problem in Banach spaces

This paper deals with a regularity assumption for the mathematical programming problem in Banach spaces. The attractive feature of our constraint qualification is the fact that it can be considered as a condition on the active part only of the constraint, and that it is preserved under small perturbations. Moreover, we show that our condition is almost equivalent to the existence of a non-empty and weakly compact set of Lagrange multipliers. The main step in the proof of our results is a generalization of the open mapping theorem.The early parts of this article result from fruitful correspondence with S. Kurcyusz, who died tragically in 1978. This paper is dedicated to his memory.     

A mathematical programming model for integrating production and procurement transport decisions

In this paper, we propose a new mathematical programming model for integrating production and procurement transport planning decisions in manufacturing systems in a unique optimization model. This problem was introduced conceptually and dubbed as MRP IV by Díaz-Madroñero et al. (2012) to extend the current MRP (material requirement planning) systems. This proposal simultaneously considers material, production resources capacities and procurement transport planning decisions with different shipping modes (such as full-truckload, less-than-truckload and milk-run) in the supply chain to avoid suboptimal results, which are usually generated due to sequential and independent plans. We considered an industrial automobile company to validate the proposed model using real world data. The results obtained by the MRP IV proposed model, in terms of total planning costs and transport efficiency indicators, are better than those obtained in the current heuristic procedures followed in the company under study.     

Necessary conditions for an extremum in a mathematical programming problem

For minimization problems with equality and inequality constraints, first-and second-order necessary conditions for a local extremum are presented. These conditions apply when the constraints do not satisfy the traditional regularity assumptions. The approach is based on the concept of 2-regularity; it unites and generalizes the authors’ previous studies based on this concept.     

A multi-objective mathematical model for the industrial hazardous waste location-routing problem

Industrial hazardous waste management involves the collection, transportation, treatment, recycling and disposal of industrial hazardous materials that pose risk to their surroundings. In this paper, a new multi-objective location-routing model is developed, and implemented in the Marmara region of Turkey. The aim of the model is to help decision makers decide on locations of treatment centers utilizing different technologies, routing different types of industrial hazardous wastes to compatible treatment centers, locations of recycling centers and routing hazardous waste and waste residues to those centers, and locations of disposal centers and routing waste residues there. In the mathematical model, three criteria are considered: minimizing total cost, which includes total transportation cost of hazardous materials and waste residues and fixed cost of establishing treatment, disposal and recycling centers; minimizing total transportation risk related to the population exposure along transportation routes of hazardous materials and waste residues; and minimizing total risk for the population around treatment and disposal centers, also called site risk. A lexicographic weighted Tchebycheff formulation is developed and computed with CPLEX software to find representative efficient solutions to the problem. Data related to the Marmara region is obtained by utilizing Arcview 9.3 GIS software and Marmara region geographical database.     

A chance constraints goal programming model for the advertising planning problem

This paper presents a model which has been designed to decide the number of advertisement in different advertising media and the optimal allocation of the budget assigned to the different media. The main objective of this problem is to maximize the reach to the desired section of people for different media within their maximum allowable budget without violating the max and min number of advertisement goals. The media have been considered as different newspapers and different channels in Televisions. Here in this article the model has been formulated in such a way that the advertisement should reach to those who are suitable for the product instead of going to those section who are not considered suitable for the product as well. A chance constrained goal programming model has been designed after considering the parameter corresponding to reach for different media as random variables. The random variables in this case has been considered as values which have known mean and standard deviations. A case for an upcoming institution who are interested to advertise for its two years Post Graduate Diploma in Management (PGDM) programme to the different newspapers and television channels has been designed to illustrate the solution methodology.     

Evaluating the effect of gap size in a single function mathematical programming model for the three-group classification problem

This study examines the impact that the size of the classification gap can have on the classificatory performance of a mathematical programming based discriminant model. In mathematical programming based models that project the discriminant scores onto a line, the discriminant score of an observation may fall into the gap between adjacent group intervals; thus there is no clear cut way to determine the group in which the observation should be classified. We examine a procedure that we refer to as the split gap approach. The split gap approach is defined as a strategy of estimating the performance of a mathematical programming based model using a nonzero gap size to separate group intervals and then splitting the gap between adjacent group intervals to classify future observations. Studies that propose models with a classification gap generally do not assess the effect of the gap on the performance of the model. This paper investigates this effect. A theoretical assessment and a Monte Carlo simulation are used to determine the impact of different gap sizes on a mixed integer programming model using a single function classification model for the three-group case.     

Solving a school bus scheduling problem with integer programming

In many rural areas in Germany pupils on the way to school are a large if not the largest group of customers in public transport. If all schools start more or less at the same time then the bus companies need a high number of vehicles to serve the customer peak in the morning rush hours. In this article, we present an integer programming model for the integrated coordination of the school starting times and the public bus services. We discuss preprocessing techniques, model reformulations, and cutting planes that can be incorporated into a branch-and-cut algorithm. Computational results show that in our test counties a much lower number of buses would be sufficient if the schools start at different times.     

A mathematical model and algorithms for the aircraft hangar maintenance scheduling problem

An aircraft hangar maintenance scheduling problem is studied, motivated by the aircraft heavy maintenance conducted in a hangar operated by an independent maintenance service company. The aircraft hangar maintenance scheduling problem in such context consists of determining a maintenance schedule with minimum penalty costs in fulfilling maintenance requests, and a series of hangar parking plans aligned with the maintenance schedule through the planning period. A mixed-integer linear programming (MILP) mathematical model, integrating the interrelations between the maintenance schedule and aircraft parking layout plans, is presented at first. In the model, the variation of parking capacity of the maintenance hangar and the blocking of the aircraft rolling in and out path are considered. Secondly, the model is enhanced by narrowing down the domain of the time-related decision variables to the possible rolling in and out operations time of each maintenance request. Thirdly, to obtain good quality feasible solutions for large scale instances, a rolling horizon approach incorporating the enhanced mathematical model is presented. The results of computational experiments are reported, showing: (i) the effectiveness of the event-based discrete time MILP model and (ii) the scalability of the rolling horizon approach that is able to provide good feasible solutions for large size instances covering a long planning period.     

A mathematical programming model for the long term development of an off-shore gas field

Commercial mathematical programming codes can solve practical problems in combinatorial optimization. But the formulation is not always straightforward, partly because of nonlinearities and partly because the code may work badly if the LP relaxation of the model does not provide a good guide to the integer solution.This paper describes a model that has been developed by Scicon and BP Petroleum Development (UK) Ltd., and used to study alternative developments of an off-shore gas field. The formulae defining gas production rates, and the capacities of compressors and pipelines are nonlinear, and their transformation to a form suitable for a general mathematical programming code is discussed in detail.Computational experience is summarized.     

The Euclidean Steiner tree problem in Rn: A mathematical programming formulation

A nonconvex mixed-integer programming formulation for the Euclidean Steiner Tree Problem (ESTP) in Rⁿ is presented. After obtaining separability between integer and continuous variables in the objective function, a Lagrange dual program is proposed. To solve this dual problem (and obtaining a lower bound for ESTP) we use subgradient techniques. In order to evaluate a subgradient at each iteration we have to solve three optimization problems, two in polynomial time, and one is a special convex nondifferentiable programming problem. This revised version was published online in June 2006 with corrections to the Cover Date.