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Lesson 2:Optimisation Techniques

Lesson 2:Optimisation Techniques. Mathematics of ‘Optimization’. ‘Optimization’  a decision maker wishes to either MAXimize or MINimize a goal (i.e. objective function) A manager’s goal is to increase profit. Profits increase take place by a. Maximising output b. Minimise cost.

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Lesson 2:Optimisation Techniques

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  1. Lesson 2:Optimisation Techniques

  2. Mathematics of ‘Optimization’ ‘Optimization’  a decision maker wishes to either MAXimize or MINimize a goal (i.e. objective function) A manager’s goal is to increase profit. Profits increase take place by a. Maximising output b. Minimise cost

  3. Optimisation • Ways of expressing economic relationships • total, average, marginal • Process of optimisation – Total and marginal approach • Differentiation • Optimisation with calculus • New Management tools for optimisation

  4. Economic relationship Economic relationships can be expressed in equational, tabular or graphical form. Expressing an economic relationship in equational form is very useful because it allows use of techniques of differential calculus to determine optimal behavior of firm. We use concepts of total, average and marginal for optimisation. TR = 100Q – 10 Q2 If TC=150Q-3Q2+.25Q3, find MC and AC.

  5. Relation between total, average and marginal

  6. Total, Average, andMarginal Cost

  7. OPTIMISATION Profit Maximisation • Π = TR – TC (PQ – CQ) • MC = MR (mc should cut mr from below) • What output should be produced • What price should be charged • What is minimum cost of production For a, b, we need to know demand function For c, we need to know cost function

  8. Profit Maximization

  9. Marginal Analysis • Analysis of ‘marginal’ costs and ‘marginal’ benefits due to a change • Marginal = additional or incremental • Costs and benefits that are constant (i.e. fixed, don’t change) are excluded from the analysis • Changes occurring at ‘the margin’ are all that matter

  10. Marginal Analysis(Examples)

  11. Optimal level of output • a. Mπ= 0 • b. MR = MC Once output has been determined, the firm’s optimal price is found from price equation and profit can be estimated accordingly

  12. Constant Rule Power Function Rule Sum-Difference Rule Product Rule Chain Rule Quotient Rule If y = f(x) = C then dy/dx = 0 If y = f(x) = Cxn then dy/dx = nCxn-1 If y = f(x) +g(x) then dy/dx = df(x)/dx + dg(x)/dx If y = f(x)·g(x) then dy/dx = df(x)/dx·g(x) + f(x)·dg(x)/dx If y = f(z) and z = g(x) then dy/dx = dy/dz·dz/dx If y = f(x)/g(x) then Rules of Differentiation

  13. Differential Analysis • Optimisation analysis is conducted more efficiently with differential analysis. For a function to be at its maximum or minimum, the derivative of the function must be zero. • To distinguish between maximum and minimum, use the second derivative. If the 2nd derivative is positive, it is minimum point. If 2nd derivative is negative, it is maximum point

  14. Optimization With Calculus Find X such that dY/dX = 0 Second derivative rules: If d2Y/dX2 > 0, then X is a minimum. If d2Y/dX2 < 0, then X is a maximum.

  15. Partial derivative • For a function having 2 or more independent variables y= f (x, z), the partial derivative dy/dx is the slope of the relationship between y and x, assuming z to be constant • Optimising a multivariate function requires setting each partial derivative equal to zero and then solving the resulting system of equations simultaneously for values of each independent variable

  16. Constrained Optimisation • A constrained optimisation problem may be solved by first solving the constraint equation for one of the decision variables, and then substituting the expression for this variable into the objective function that the firm seeks to maximise or minimise

  17. Probability • A probability distribution lists the possible outcomes of experiment and prob associated with each outcome • Prob of outcome is 0< P (Xi)<1 • Sum of all prob =1

  18. Common Math Functions Used in Economics

  19. Linear Demand Function MR Revenue under Linear Demand P or AR • Total Revenue • Marginal Revenue • ARC Marginal Revenue

  20. Key concepts • The value of average function at any point is the slope of a ray drawn from origin to total function at that point • The value of marginal function at any point is the slope of a ray drawn tangent to total function at that point • Marginal function will intersect average function at either minimum or maximum point of average function

  21. Contd… • If marginal function is positive, the total function will be increasing. If marginal function is negative, total function will be decreasing • The total function reaches a maximum or minimum when marginal function equals zero

  22. Optimisation techniques • Benchmarking • TQM • Reengineering • Learning Objectives

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