Mathematical Programming Approaches

Abstract

Mathematical programming is one of the most important techniques available for quantitative decision-making. The general purpose of mathematical programming is finding an optimal solution for allocation of limited resources to perform competing activities. The optimality is defined with respect to important performance evaluation criteria, such as cost, time, and profit. Mathematical programming uses a compact mathematical model for describing the problem of concern. The solution is searched among all feasible alternatives. The search is executed in an intelligent manner, allowing the evaluation of problems with a large number of feasible solutions.Mathematical programming finds many applications in supply chain management, at all decision-making levels. It is also widely used for supply chain configuration purposes. Out of several classes of mathematical programming models, mixed-integer programming models are used most frequently. Other types of models, such as stochastic and multi-objective programming models, are also emerging to handle more complex supply chain configuration problems. Although these models are often more appropriate, computational complexity remains an important issue in the application of mathematical programming models for supply chain configuration.This chapter describes application of mathematical programming for supply chain configuration. The general overview is given in Sect. 8.2. It is followed by a description of generic supply chain configuration mixed-integer programming model in Sect. 8.3. This model is based on the data model presented in Chap. 7. Computational approaches for solving problems of large size are also discussed along with typical modifications of the generic model, especially, concerning global factors. Section 8.4 outlines the application of other classes of mathematical programming models. In Sect. 8.5, the generic optimization model is used to optimize the SCC Bike's supply chain configuration. Section 8.6 details a model

A Review of Models and Algorithms for Surface-Underground Mining Options and Transitions Optimization: Some Lessons Learnt and the Way Forward

“…model involving more than one objective function is deployed simultaneously. The main advantages of mathematical programming models (MPMs) are: (a) comparatively simple with high approximations of complicated problems, and (b) ability to search the feasible solution spaces among competing variables and alternatives [82]. Common MP techniques are stochastic programming, deterministic programming, dynamic programming, LP, nonlinear programming, IP, and MILP.…”

“…Stochastic Programming Stochastic programming is a special case of programming in which some of the constraints or parameters depend on random variables. This type of programming is used to solve problems that involve uncertainty and allow stochastic variables to be accounted for [82].…”

“…Appendix A.4.3. Dynamic Programming Dynamic programming is a form of mathematical programming used to solve multistage complicated problems [82]. A large-scale problem is simplified by breaking it down into simpler sub-problems in a nested recursive manner as opposed to deterministic programming.…”

“…Nonparametric and simulation-based optimization techniques have been used to explicitly solve models with nonlinear constraints. However, computational feasibility have been a major disadvantage in the development and implementation of MPMs [82]. Appendix A.4.6.…”