Select Page

# Author: espin086

## Simplex Algorithm and How Dating Web Sites Match Singles

Mathematical Background: The simplex method was created by George Dantzig (1914-2005)  who was Professor Emeritus of Transportation Sciences and Professor of Operations Research and Computer Science at Stanford University.  George Dantzig created the simplex method as an algorithm for solving a system of linear equations.  The algorithm is used in linear programming to find optimum solutions to these equations.  The equations typically consist of one objective function which you are trying to minimize (i.e. cost) or the dual problem which would consist of an equation your are trying to maximize(i.e. profit). In addititon to the objective function you will have some sort of constraints which will limit your optimum solution.  These constraints could be in the form of storage capacity, capital budgets, time constraints, labor expenses, substitutability of inputs and a host of other factors can play a part in defining the set of constraints. The original applications to the simplex method were to linear programming.  Linear programming was used during WWII was as a way of minimizing cost to the army and increase losses to the enemy through superior planning and utilization of resources. Examples of the uses of the simplex method and linear programming include the transportation problem where the algorithm minimizes the cost of shipping between n number of warehouses and m number of destinations.  The diet problem is another application where the nutritional needs of an army are taken into account as we try and minimize...

## Lagrangian and Production Maximization with Linear Budgetary Constraints

Mathematical Background: Multivariable calculus is merely an extension of single-variable calculus.  The multivariate approach allows for the use of more sophisticated models where algebra is insufficient.   Multivariable calculus is used in many fields and disciplines including engineering, physics, social sciences and many others where there is a deterministic system that is the object of mathematical examination.  The many applications of calculus include computer science, medicine, statistics, demography, electromagnetism, and economics.  Calculus has a long history-from the calculations of the volume of pyramids in ancient Egypt to the rate of contagion of the recent Swine Flu.  In particular, any discipline which has change (derivative) or that has an accumulated buildup (integral) can be subject to analysis with calculus. Economics is one of those disciplines that uses calculus extensively to develop its theory of production.   The basic model in economics is involves two inputs, labor and capital, where their individual properties are homogenous.   These two inputs form the independent varibles in a production function which outputs the total quantity produced of a given product.  There is a curve called an iso-quant curve which describes a fixed quantity to be produced and all the possible combinations of labor and capital that can achieve this output.  There is a cost function that is the sum of the inputs multiplied by the price of inputs to equal the total cost.  There is also an...

## Linear Programming and Production Schedule Optimization

Mathematical Background: Linear programming problems can be used to solve many problems in transportation, production, and commodity pricing.  Variations of linear programming problems can arise when one wants to answer questions of maximization or minimization, but the overall techniques is homogenous among most variations of the problems.  Non-linear programming problems, such as the Cobb-Douglas Production Function maximization can be handled with multivariable calculus, but many interesting problems such as the assigment model or the transportation model are discrete in nature.  As the number of constraints approaches infinity then the linear-programming problem becomes non-linear and subject to these calculus techniques.  In this entry the continuity of the function will be replaced with finite constraints so that only the methods used in linear programming will be useful, but it is important to note that in the limit there is a convergence between the linear and non-linear programming methods. Linear Programming is a technique for optimizing a linear objective function subject to a set of linear constraints.  Typically the set of linear constraints are in the form of equalities and inequalites which converm a convex polyhedron.  This convex set determines the feasible solution region in cases where the problem has a feasible solution and is properly defined.  The method for solving linear optimization problems within this feasible convex set is called the Simplex Method. The simplex method was created by George Dantzing who...

## Multivariate Regression Analysis and Choosing Home Improvements for Profit

Mathematical Background: Regression Analysis is a mathematical technique for estimating the value of parameters in a linear equation for the creation of extremely useful mathematical models that quantify the relationship between variables. The term regression comes from Sir Fransis Galton for the term previously described by the great mathematician Fredrik Gauss and Legendre as “least square”.  Legenre and Gauss both applied regression analysis to the observation of planetary movements to make inferences and predictions about their future positions in the sky. Regression analysis is one of the most important mathematical tools ever discovered.  The applications to linear regression include finding the connection between cigarette smoking and lung cancer, quantifying systemic risk of a stock investment, determining the impact of breast feeding on psychological health and the impacts of different home improvements on the potential resale value of a home.  Regression has even been used to forecast the expected increase in crime as a result of an increase in the price of heroin.  These analyses can better help society plan for the future, determine the toxicity of products, and help individuals generate income through real estate. Regression analysis is also misused and abused in data analysis so there must be a series of hypothesis testing conducting after a regression has been calculated.  Hypothesis testing is a key feature in determining the statistical significance of the independent variable parameters.  The topic of...

## Poisson Probability Distribution Functions and Call Waiting Times

Mathematical Background: The Poisson Probability Cumulative Density function is useful in many applications where the mean of a certain activity is known and questions are arised regarding the probability of certain ranges of values for the stochastic activity.  There are many examples of the applications of this distribution including product warranty ranges, productivity bonus expense estimations, and the probability of  a certain range of items returned for a refund for a manufacturing organization. One important question that a call center might want to ask is the number of minutes that it keeps its customers waiting for a customer service representative.  This measurement is important to customer satisfaction and cost reduction, both of which are components to the companies overall profitability.  Overestimating the waiting time will result in the company hiring additional employees and incurring an additional cost which might not be worth the reduction in waiting time.  A similar problem arises when the company underestimates the time spent by customers waiting-the savings in wage expenses are not offset by the dissatisfaction of customers having to wait a long time to be heard and consequently increasing the firms cost by initiating a chargeback.  Chargebacks are credits issued to customers by their credit card issuing bank as a result of a claim by the customer as to the validity of charges present on their credit cards.  High chargeback ratios places the contractual...