Linear Programming Approaches for Multiple-Class Discriminant and Classification Analysis

A mixed integer programming model is proposed for multiple-class discriminant and classification analysis. When multiple discriminant functions, one for each class, are constructed with the mixed integer programming model, the number of misclassified observations in the sample is minimized. This model is an extension of the linear programming models for multiple-class discriminant analysis but may be considered as a generalization of mixed integer programming formulations for two-class classification analysis. Properties of the model are studied. The model is immune from any difficulties of many mathematical programming formulations for two-class classification analysis, such as nonexistence of optimal solutions, improper solutions, and instability under linear data transformation. In addition, meaningful discriminant functions can be generated under conditions where other techniques fail. Examples are provided. Results on publically accessible datasets show that this model is very effective in generating powerful discriminant functions.

In this chapter, the term discriminant analysis refers to descriptive group separation using (linear) discriminant functions to characterize the differences between two or more groups. The goals of descriptive discriminant analysis include identifying the relative contribution of the p variables to separation of the groups and finding the optimal plane on which the points can be projected to best illustrate the configuration of the groups. Prediction or allocation of observations to groups (predictive discriminant analysis) is referred to in this book as classification analysis and is discussed in Chapter 9. For two groups, the discriminant function can be calculated using matrix manipulation or by multiple regression with dummy grouping variables. For the several-group case, the discriminant function coefficient vectors are obtained using eigenvectors. In either case, the coefficients can be standardized to better show the relative contribution of the variables to separation of the groups. The discriminant functions can be tested for significance. Various approaches are given to interpreting the discriminant functions or determining the contribution of each variable. The techniques in the chapter are well illustrated by examples using real data. The problems at the end of the chapter provide derivations and further numerical illustrations.

A cutting plane algorithm for the site layout planning problem with travel barriers

In this paper, we propose a mixed integer programming (MIP) model to solve the experimental courses scheduling problem. And we use the experimental courses historical data in our information systems to solve the MIP model. The results of the numerical example demonstrate that the proposed model can reach an optimal solution in a short period of time. The MIP model can provide the decision-making support for the experimental courses arrangement.

This paper considers the hybrid flow shop scheduling problem, where jobs are processed in <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</i> stages with the same route of the stage. Each stage has identical parallel machines for processing jobs. Some mathematical programming formulations and lower bound calculations have been proposed in the literature for such cases. Nevertheless, there is a lack of complete comparisons of these mathematical programming formulations and lower bounds in the hybrid flow shop literature. In this paper, we propose a new mixed integer programming model and two new lower bounds based on the bin-packing concept for the considered problem. To evaluate the proposed model, two sets of small and small-to-medium problems are used to compare our model with the existing models. Moreover, two propositions are discussed for lower bounds. The experimental results show that the proposed mixed integer programming model efficiently found optimal solutions because it needs a smaller number of binary variables and a smaller number of constraints, and the proposed lower bound can also serve as a strong indicator to evaluate the distances between the solutions obtained by heuristic algorithms and the optimal solution.

Factor analysis, classification and regression tree analysis (CART), discriminant analysis and classification analysis were applied to study land cover in Wyoming, to explore: 1) how environmental variables are related to one another; 2) whether land cover types (forest, grass, shrub and un-vegetated) are differentiated from one another in term of abiotic conditions; and 3) how to predicate vegetation covers based on environmental variables. A factor analysis indicated that environmental variables were characterized by three dominant conditions: 1) harsh condition for survival, 2) suitable condition during growing season, and 3) harsh condition caused by high altitude. A discriminant analysis indicated that: 1) forests and grasses required wet climate with relatively cool summers, while shrub could bear drought; 2) quite different to forests, grasses required dry winters and relatively wet and warm summers. CART and classification analysis both indicated that, by using Precipitation in July, Maximum Annual Temperature and Average Annual Precipitation, we had a fair accuracy in predicting vegetation covers in Wyoming. Keywords: Factor analysis, discriminant analysis, classification and regression tree, land cover, Wyoming .

Efficient operational planning in district heating systems (DHSs) is essential for minimizing operating costs and maximizing energy efficiency. However, since practitioners must determine future production plans under unknown future demands and costs in real-world energy systems, it is challenging to solve the production planning problems of DHSs. In this paper, we propose a multi-head attention-based TimesNet (MATN) in which a transformer decoder is incorporated that operates solely on a 24 h lookback window without requiring any future information. Specifically, the model is trained in an end-to-end manner, for which the training dataset was built by solving a mixed integer programming (MIP) model. Experimental results demonstrate that the proposed MATN model significantly outperforms baseline deep learning-based methods. A qualitative analysis of the hourly production plans further indicates that MATN generates robust operational plans that mimic those generated by an MIP model, which suggests the effectiveness of the proposed approach in terms of economic efficiency and operational stability without depending on future information.

In this paper, we propose a mixed integer programming (MIP) model for the Chicken Flock Sizing, Allocation and Scheduling Problem (CFSASP), which is an important planning problem in the broiler production supply chain. To solve the CFSASP efficiently, two variants of rolling horizon heuristics (RHHs) have been developed and applied on the case of a Norwegian broiler production company. Computational results show that the RHHs successfully obtain high-quality solutions within a reasonable time. The value of optimisation is verified through comparison with the case company's plans, where the solutions from optimisation outperforms the current solutions. Sensitivity analyses are also conducted to provide managerial insights regarding certain strategic decisions, such as how many and which days to use for hatching of chickens. Due to the promising results, the case company is now implementing an optimisation-based decision support system based on the MIP model and solution methods shown in this paper.

In apparel manufacturing, cut pieces produced by the cutting process are a key input work-in-process (WIP) to the bottleneck sewing process. However, an excessive inventory of cut pieces not only requires extra storage space but also creates difficulties with shop floor control. This study determines the ideal cutting times of fabric lays by minimising the cut piece inventory under the constraint that sufficient cut pieces are produced in time to satisfy the bottleneck usage. This study proposes a mixed integer programming (MIP) model and a heuristic method to solve the problem. The solutions for most test instances of the problem that are calculated by the heuristic method are not as good as those by the MIP solver; however, the heuristic method computation times are negligible. The experiments show that the MIP model’s efficiency significantly improves by starting from an initial solution provided by the proposed heuristic method.

Integrated optimization of traffic signal timings and vehicle trajectories considering mandatory lane-changing at isolated intersections

Lattice-based algorithms for number partitioning in the hard phase

The flying sidekick traveling salesman problem with stochastic travel time: A reinforcement learning approach

Minimizing total tardiness on a single machine with controllable processing times

Using discriminant function analysis to estimate the impact of a therapeutic intervention in burn care

Although powerful for the resolution of the classification problems, the major disadvantage of the parametric procedures (Linear Discriminant Function (LDF), Quadratic Discriminant Function (QDF) and Logistic Regression) is their requirement of certain assumptions. These assumptions are normality, equality of variance-covariances matrix, the absence of outliers, etc. To fill this insufficiency, several researchers such as Freed and Glover in 1980, were interested in the resolution of the classification problems via linear programming approaches. Nevertheless, the two above mentioned approaches suppose that the variables (or attributes) are measured with certainty. However, in an increasingly complex environment, these variables can be imprecise, qualitative or linguistic. In such a case, fuzzy set theory seems to be the convenient tool to fill this insufficiency. Thus, we proposed a new approach, which consists in solving the classification problems via Fuzzy Linear Programming Models (FLPM).