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Applied Math
Linear Programming
Quan Meth Exam 1 Miller
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Fredrick Taylor
--Father of management science
--Brought science into the workplace
--Major Flaw
-Management science wasnt enough
-HR, Org B, Opps Management
What is Soldiering
--Group work causes best performers to perform worse.
Describe Management Science
Def: Scientific approach to solving management problems.
-Also known as Operation Research, Decision Sciences, Quan Methods.
Management Science Outlook/Philosophy
--Simple
--Logical
Management Science Process
1.Observation
2.Problem Definition
3.Model Construction
4.Solution
5.Implementation
-FEEDBACK on 2-5
Discuss Management Science Steps.
-Observation
--Identification of a problem that exists (or may occur soon) in a system or organization.
Discuss Management Science Steps.
-Definition of Problem
--Problem must be clearly and consistently defined, showing its boundaries and interactions with objectives of the organization.
Discuss Management Science Steps.
-Model Construction
--Development of the functional mathematical relationships that describe decision variables, objective functions and constraints of the problem.
Discuss Management Science Steps.
-Model Solution
--Models solved using management science techniques.
Discuss Management Science Steps.
-Model Implementation
--Actual use of the model or its solution.
Probablistic Model
--Model parameters contain uncertainty
-Based on averages and distributions
--Provide solutions which are statistical averages
Deterministic Model
--Assume parameters are certain
--Provide one solution (Optimal)
Describe Business Analytics
--Uses large amounts of data with management science techniques to help managers make decisions.
--Classifies Models
-Descriptive, Predictive, Prescriptive
--Brings together information technology, statistics, management science, and mathematical modeling.
--Big Data
--Data Mining
Describe Break-Even Analysis
--Used to determine the number of units of a product to sell or produce (i.e. Volume) that will equate total revenue with total cost.
--The volume at which total revenue = total cost is called break-even point.
--Profit at break-even point is zero.
--Mathematical, Graphical, Computer Solutions
-Test vs. Problems.
Components of Break-Even Analysis
--Fixed Costs (Cf)
-Costs that remain constant regardless of number of units produced.
--Variable Costs (Cv)
-Unit production cost of product.
--Total Variable cost (VCv)
-Function of volume (V) and unit variable cost.
--Total Cost (TC) total fixed cost plus total variable cost.
- TC=Cf+VCv
--Profit (Z)
-Difference between total revenue (TR=VP) (P=Unit Price) and total cost.
- Z=TR-TC
Z=VP-(Cf+VCv)
Z=VP-Cf-VCv
Break Even Point
Def: Volume at which total revenue = total cost and profit is zero.
--TR=Tc
VP=Cf+VCv
Vp-VCv=Cf
V(P-Cv)=Cf
V=Cf/(P-Cv)(BE Point)
-BE point as percent of capacity =V/Capacity=%
Define Linear Programming
Def: Analytical technique in which linear algebraic relationships represent a firm's decisions, given a business objective, and resource constraints.
--Objectives of business decisions frequently involve maximizing profit or minimizing costs.
Linear Programming Steps
1.Identify problem as solvable by linear programming.
2.Formulate a mathematical model of the unstructured problem.
3.Solve the model
4.Implementation
Linear Programming Components
--Decision Variables
--Objective Function
--Constraints
--Parameters
Decision Variables
Def: Mathematical symbols representing levels of activity of a firm.
Objective Function
Def: Linear mathematical relationship describing an objective of the firm, in terms of decision variables.
-This function is to be maximized or minimized.
Constraints
--Requirements or restrictions placed on the firm by the operation environment, stated in linear relationships of the decision variables.
Parameters
--Numerical coefficients and constraints used in the objective function and constraints.
Steps for forming Linear Programming
1.Clearly define the decision variables.
2.Construct the objective function
3.Formulate the constraints
Feasible solution
Def: Does not violate any of the constraints.
Infeasible solution
Def: Violates atleast one of the constraints.
Graphical Solution of Linear Programming Models
--Limited to linear programming models containing only two decision variables (can be used with 3 but only with great difficulty).
Slack
Def: Amount of a resource that is not used.
Slack Variables
--Standard form requires that all constraints be in the form of equations (equalities)
--A slack variable is added to a <_ constraint (weak inequality) to convert it to an equation (=)
--A slack variable typically represents an unused resource.
--A slack variable contributes nothing to the objective function value.
Irregular Types of Linear Programming Problemm
--For some linear programming models, the general rules do not apply.
--Special types of problems include those with:
-Multiple optimal solutions
-Infeasible solutions
-Unbounded solutions
Characteristics of Linear Programming
--Requires a decision - a choice amongst alternative courses of action.
--The decision is represented in the model by Decision Variables.
--The problem encompasses a goal, expressed as an objective function, that the decision maker wants to achieve.
--Constraints exist that limit the extent of achievement of the objective.
--The objective and constraints must be defined by linear mathematical functional relationships.
Properties of Linear Programming Models
-Proportionality
--The rate of change (slope) of the objective function and constraint equations is constant. (Linear Relationships-Not Non-Linear Programming)
--Means that we are dealing with linear relationships, all of the lines on the graph are straight, not any curves.
Properties of Linear Programming Models
-Additivity
--Terms in the objective function and constraint equations must be additive. (Cannot have multiplicative or divisible equations)
--Means that there cannot be multiplicative equations.
Properties of Linear Programming Models
-Divisibility
--Decision variables can take on any fractional values and are therefore continuous as opposed to integer in nature. (Not Integer Programming)
--Means solution can be any fractional value.
Properties of Linear Programming Models
--Certainty
--Values of all the model parameters are assumed to be known with certainty. (Non-probabilistic)
--LP is deterministic, there is one answer and where all model parameters are known.
Linear Programming Problem
-Standard Form
--Requires all variables in the constraint equations to appear on the left of the inequality (Or equality) and all numeric values to be on the right hand side.
Sensitivity Analysis
--Determines the effect on the optimal solution of changes in parameter values of the objective function and constraint equations.
--Changes may be reactions to anticipated uncertainties in the parameters or to new or changed information concerning the model.
Sensitivity Range
--For an objective function coefficient is the range of values over which the current optimal solution point will remain optimal.
--For the Xi coefficient is designated as Ci.
Range Table
--Can only use for two types of changes:
-Parameter Objective Function
-Resource availability
--Cannot use for:
-Adding Decision Variable
-Adding Constraint
-Changing parameter constraint
Changes in Constraint Quantity Values Sensitivity Range
--The sensitivity range for a right hand side value is the range of values over which the quantity's value can change without changing the solution variable mix, including the slack variables.
Other Forms of Sensitivity Analysis
-Changing individual constraint parameters.
-Adding new constraints
-Adding new variables
--Cannot use range table to determine impact of these changes.
Types Of Integer Programming Models
--Total Integer Model
--0-1 Integer Model
--Mixed Integer Model
Total Integer Model
--All decision variables required to have integer solution values.
--The decision must involve whole numbers.
0-1 Integer Model
--All decision variables required to have integer values of zero or one.
Mixed Integer Model
--Some of the decision variables (but not all) required to have integer values.
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Verified questions
ADVANCED MATH
Assume that an average of 125 packets per second of information arrive to a router and that it takes an average of .002 second to process each packet. Assuming exponential interarrival and service times, answer the following questions. a. What is the average number of packets waiting for entry into the router? b. What is the probability that 10 or more packets are present?
ADVANCED MATH
Eli Daisy uses chemicals 1 and 2 to produce two drugs. Drug 1 must be at least 70% chemical 1, and drug 2 must be at least 60% chemical 2. Up to 40 oz of drug 1 can be sold at $6 per oz; up to 30 oz of drug 2 can be sold at$5 per oz. Up to 45 oz of chemical 1 can be purchased at $6 per oz, and up to 40 oz of chemical 2 can be purchased at$4 per oz. Formulate an LP that can be used to maximize Daisy’s profits.
ADVANCED MATH
Howard Whose has left an estate of $200,000 to support his three ex-wives. Unfortunately, Howard’s attorney has determined that each ex-wife needs the following amount of money to take care of Howard’s children: wife 1—$ 100,000; wife 2—$200,000; wife 3—$300,000. Howard’s attorney must determine how to divide the money among the three wives. He defines the value of a coalition S of ex-wives to be the maximum amount of money left for the ex-wives in S after all ex-wives not in S receive what they need. Using this definition, construct a characteristic function for this problem. Then determine the core and Shapley value for this game.
ADVANCED MATH
a) Verify that the diagonals of the rectangle with vertices J(-2, 1), K(2, 3), L(4, -1), and M(0, -3) bisect each other at right angles. b) Do all rectangles have this property? c) What can you conclude about the lengths of the sides of JKLM? Explain your reasoning.