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Linear Programming
Published in Anindya Ghosh, Prithwiraj Mal, Abhijit Majumdar, Advanced Optimization and Decision-Making Techniques in Textile Manufacturing, 2019
Anindya Ghosh, Prithwiraj Mal, Abhijit Majumdar
LP methods are some of the most popular, versatile, and commonly used approaches that provide optimal solutions to the problems with linear objective and constraint functions. This chapter presents a detailed application of graphical and simplex methods of solving linear programs with suitable examples related to textile manufacturing, along with coding in MATLAB® language. In the graphical method, initially a feasible region is identified in a two-dimensional plane, after satisfying all of the constraints simultaneously. Then, by a trial-and-error method, a point is located in the feasible region whose coordinate gives the optimum value. Though the graphical method is easy to understand, it can only be used when the number of variables in a linear problem is limited to two. Simplex methods are found to be very efficient in solving LP problems of any magnitude. The simplex method begins with a basic feasible solution and at each iteration it projects an improved solution over the earlier step. A solution is considered optimum when there is no further improvement. Simplex methods require formulation of the LP in a standard form in which all the inequality constraints are converted into equality constraints by using slack, surplus, or artificial variables. Then key operations at each iteration are performed to obtain the optimal solution. Published works on application of LP in textile manufacturing are also illustrated in this chapter.
Interior Point Methods
Published in James A. Momoh, Electric Power System Applications of Optimization, 2017
Consider the following problem. Maximize z = 2x1 + 5x2 + 7x3Subject tox1 + 2x2 + 3x3 = 6xj ≥ 0.Graph the feasible region.Find the gradient of the objective function and then find the projected gradient onto the feasible region.Starting from initial trial solution (1, 1, 1) perform two iterations of the IP algorithm.Perform eight additional iterations.
Mathematical methods for structural synthesis
Published in József Farkas, Károly Jármai, Analysis and Optimum Design of Metal Structures, 2020
It is very important to determine, under what condition a local optimum is also a global one. It depends on the form of the feasible region, determined by the constraints. For a convex region the local optimum is a global one, otherwise there are several local optima (Figs 6.3a and b). Convex is a region, if between any two interior points all points are also interior, otherwise the region is non-convex.
A generalized Benders decomposition approach for the mean-standard deviation shortest path problem
Published in Transportation Letters, 2022
Let and represent the optimal solutions of problems and , respectively. Figure 2 displays the search region of the Benders master problem consisting of the infeasible region, feasible region, and dominated region. The constraints (21) and (22) define the feasible region.