Mixed integer linear programming models for optimal crop selection

In this paper, we propose the modeling of a real-case problem where a farmer has to optimize the use of his/her land by selecting the best mix of crops to cultivate. Complexity of the problem is due to the several factors that have to be considered simultaneously. These include the market prices variability of harvested products, the specific resource requests for each crop, the restrictions caused by limited machines availability, and the timing of operations required to complete each crop cultivation. We provide two different mathematical formulations for the analyzed problem. The first one represents a natural integer programming formulation looking for the crop-mix that maximizes the farmer’s expected profit measured as the difference between revenues obtained by selling the harvested products and the production costs. Since the revenue of each crop depends on the price as quoted at the exchange market and the yield per hectare of harvested product, we define it as a random variable. Then, the second model uses the maximization of the Conditional Value-at-Risk (CVaR) as objective function and looks for the crop-mix that allows to maximize the average expected profit under a predefined quantile of worst realizations. To test and compare the proposed models with the cultivation choice made by the farmer, we use Italian historical data represented by monthly returns of different crops over a time period of 16 years. Computational results emphasize the advantage of using the CVaR model for a risk-averse farmer and provide interesting insights for farmers involved in similar problems.     

多周期混合整数规划购油模型及应用

采用PIMS软件中的多周期混合整数规划技术建立炼油企业购油计划模型，使优化结果与实际购油方式相吻合；采用虚拟周期方法解决原油期末库存质量控制问题；采用滚动处理方式解决炼厂月、季原油选购计划的衔接和全局优化问题。文中还给出了多周期MIP模型技术在某炼厂中的应用以及不同方案的效益对比。     

Gamma deployment problem in grids: hardness and new integer linear programming formulation

Vehicular networks are mobile networks designed for the domain of vehicles and pedestrians. These networks are an essential component of intelligent transportation systems and have the potential to ease traffic management, lower accident rates, and offer other solutions to smart cities. One of the most challenging aspects in the design of a vehicular network is the distribution of its infrastructure units, which are called roadside units (RSUs). In this work, we tackle the gamma deployment problem that consists of deploying the minimum number of RSUs in a vehicular network in accordance with a quality of service metric called gamma deployment. This metric defines a vehicle as covered if it connects to some RSUs at least once in a given time interval during its whole trip. Then, the metric parameterizes the minimum percentage of covered vehicles necessary to make a deployment acceptable or feasible. In this paper, we prove that the decision version of the gamma deployment problem in grids is NP‐complete. Moreover, we correct the multiflow integer linear programming formulation present in the literature and introduce a new formulation based on set covering that is at least as strong as the multiflow formulation. In experiments with a commercial solver, the set covering formulation widely outperforms the multiflow formulation with respect to running time and linear programming relaxation gap.     

Improving the efficiency of a mixed integer linear programming based approach for multi-class classification problem

Data classification is one of the fundamental issues in data mining and machine learning. A great deal of effort has been done for reducing the time required to learn a classification model. In this research, a new model and algorithm is proposed to improve the work of Xu and Papageorgiou (2009). Computational comparisons on real and simulated patterns with different characteristics (including dimension, high overlap or heterogeneity in the attributes) confirm that, the improved method considerably reduces the training time in comparison to the primary model, whereas it generally maintains the accuracy. Particularly, this speed-increase is significant in the case of high overlap. In addition, the rate of increase in training time of the proposed model is much less than that of the primary model, as the set-size or the number of overlapping samples is increased.     

Regularization of a lexicographic vector problem of integer programming

A stability criterion for a vector integer linear problem of lexicographic optimization is obtained. A regularization method is proposed that allows us to reduce a possible unstable output problem to a sequence of perturbed stable equivalent problems. Translated from Kibernetika i Sistemnyi Analiz, No. 6, pp. 125–130, November–December, 1999.     

The type E simple assembly line balancing problem: A mixed integer linear programming formulation

Although the simple assembly line balancing problem (SALBP) is the topic of many studies, typically they either consider minimizing the number of stations for a given cycle time (called type one), or minimizing the cycle time for a given number of stations (called type two). Rarely, type E of the problem has been considered. In the type E, cycle time and number of stations are both decision variables, and the objective is to maximize the line efficiency. This paper presents a mixed integer linear programming formulation for the type E simple assembly line balancing problem. Moreover, to further strengthen the presented formulation, two enhancement techniques in the form of valid inequalities and auxiliary variables are proposed. As the secondary objectives of the problem, minimization of the number of stations, the cycle time, and the smoothness index are studied as well. In the case of workload smoothing, three different linearization methods are employed and compared for minimizing the smoothness index. The results of computational study on the benchmark data set demonstrate the efficacy of the improved formulation     

On reduction of the multistage problem of stochastic programming with quantile criterion to the problem of mixed integer linear programming

Consideration was given to the a priori formulation of the multistage problem of stochastic programming with a quantile criterion which is reducible to the two-stage problem. Equivalence of the two-stage problems with the quantile criterion in the a priori and a posteriori formulations was proved for the general case. The a posteriori formulation of the two-stage problem was in turn reduced to the equivalent problem of mixed integer linear programming. An example was considered.     

On a type of stability of a multicriteria integer linear programming problem in the case of a monotone norm

A multicriteria integer linear programming problem with a finite number of admissible solutions is considered. The problem consists in finding the Pareto set. Lower and upper attainable estimates of the radius of strong stability of the problem are obtained in the case when the norm in the space of solutions is arbitrary, and the norm in the criteria space is monotone. Using the Minkowski-Mahler inequality, a formula for calculating this radius is derived in the case when the Pareto set consists of a single solution. Estimates of the radius are also found in the case of the Hölder norm in the specified spaces. A class of problems is distinguished for which the radius of strong stability is infinite. As corollaries, certain results known earlier are derived. Illustrative numerical examples are also presented.     

A novel 0-1 linear integer programming model for dynamic machine-tool selection and operation allocation in a flexible manufacturing system

This paper considers a problem of dynamic machine-tool selection and operation allocation with part and tool movement policies in a flexible manufacturing system (FMS) environment. For this purpose, a novel 0-1 linear integer programming model is presented in such a way that each part and each tool can move during the production phase. It is assumed that there are a given set of tools and machines that can produce different kinds of orders (or part types). The objective of this model is to determine a machine-tool combination for each operation of the part type by minimizing some production costs, such as machining costs, setup costs, material handling costs and tool movement costs. In addition, due to the NP-hard nature of the problem, a new heuristic method based on five simple procedures (FSP) is proposed for solving the given problem, whose performance is tested on a number of randomly generated problems. The related results are compared with results obtained by a branch-and-bound method. It has been found that the proposed heuristic method gives good results in terms of objective function values and CPU times.     

Optimum integer number and position of several groups of prestressing tendons for given concrete dimensions

The following design problem is solved: Given are the concrete dimensions and the loadings (e.g. dead load plus traffic loading). The engineer can choose the number of groups of tendons he wishes to use. For each group the following data are given: the location of the cross-sections at which the tendons are anchored, the prestressing force of one tendon, which can also be variable along the length of the beam to take friction losses approximately into consideration, the maximum and minimum number of tendons, the lower and upper bounds of the zone permissible for guaranteeing sufficient concrete coverage (these bounds can also vary along the length of the beam), the smallest permissible radius of curvature, as well as a relative price. Several load combinations, under service conditions with specified allowable maximum and minimum concrete stresses are selected by the engineer. A reduction factor for creep and shrinkage is also included in the input.Determined are (a) the overall most favorable integer number of tendons in each group, and (b) the position along the length of the beams, such that the stress margin that results in every section and under the stress conditions formulated under the most unfavorable load positions shall be a maximum.The optimization algorithm employs in part a linear program. Methods to eliminate redundant constraints on the level of the cross-sections and the elements are discussed.The procedure is illustrated with a highway bridge (deck on raking legs), using three groups of tendons.     

Stand-alone renewable power system scheduling for a green data center using integer linear programming

Journal of Scheduling - In light of the rapid growth of data centers around the world and their huge energy consumption, several researchers have focused on the task scheduling and resource...     

A parametric analysis of linear integer programming problems

Parametric programming may be utilized to obtain restrictions to linear programming relaxations to linear integer programming problems. The purpose of such approaches is to reduce the size of the subproblems that appear subsequent to solving the relaxed linear programming problem. The optimal dual variables to the linear programming problem provide information regarding the difference between the continuous linear programming optimal objective function value and that of the integer linear programming objective function value.     

Stability radius of a vector integer linear programming problem: case of a regular norm in the space of criteria

A multicriteria integer linear programming problem of finding a Pareto set is considered. The set of feasible solutions is supposed to be finite. The lower and upper achievable bounds for the radius of stability are obtained using a stability criterion and the Minkowski–Mahler inequality and assuming that the norm is arbitrary in the space of solutions and is monotone in the space of criteria. Bounds for the radius of stability in spaces with the Holder metric are given in corollaries.