### ! OPTIMIZATION, General Introduction, Notes and Techniques

Optimization means choosing the best one from a set of alternatives according to defined objective criteria. Mathematical techniques are used to select the variable values that give the maximum or minimum value of the objective function. The objective function describes the essential characteristics of the alternatives. The limits of the variable values for each alternative can be expressed as constraints on the range of values that may be used in the objective function. Maxima or minima criteria are chosen by the qualitative nature of the objectives. E.g. costs are minimized, profits maximized.

The alternatives may be described by continuous or discrete variables. Many situations consist of choices between discrete courses of action. Discrete problems may be combinatorial and large ones may become intractable. Continuous variables may be needed to describe situations that can produce a large number of alternatives by mixing various proportions. These types of problems are tractable if the variables can be described by continuous functions.

An Optimization process selects values of independent variables which result in the maximum (or minimum) value of one or more dependent variable(s) of a value or measure of merit relationship. The relationship or 'objective function' is usually written as g(X), where g is a function of X

The independent variables are usually subject to a number of constraints in the form of other relationships and the whole group is often written as h(X). Note in this context h may be an array of functions, equations or inequalities.

X is a vector or array of independent variables which describes the process. The optimum (or optima) are groups of values of X which satisfy the optimal conditions of the objective function g and the constraints h.

The most frequently used techniques for determining the optimum value of a mathematical function are: (ref dr 10652):

` a. differential calculus               e. classical matrix method`
` b. search techniques                   f. variational calculus`
` c. direct method                       g. Bellman's Dynamic programming`
` d. linear & non linear programming     h. Pontryagin's maximum principle`

Accounting systems tend to treat costs as fixed and variable. i.e. fixed costs are not sensitive to quantity of activity while variable are directly related. These definitions lead to non linear unit costs and choosing an 'economic quantity' by 'minimizing cost'. Even if the objective is to 'maximize profits' and revenues are described by Quantity x Price relationships, the costs remain non linear. Quite often the essential problem can be made linear if the fixed costs essentially remain constant over the range of alternatives. The problem becomes one of optimizing 'the differences'.

Another technique is to choose the cost items that are fixed and ones that tend to increase per unit. This is sometimes stated as 'Costs that vary inversely with quantity' vs 'Costs that vary directly with quantity'

LOTQ is a function for estimating inventory, order and production quantity problems. The fixed cost is the amount to place an order, change tooling, bring in equipment for an operation, etc. which is unrelated to quantity.

The variable costs are those of storage, interest on inventory, etc. which are related to the maximum amount held at any one. LOTQ assumes that inventory is averaged and that production and consumption rates are uniform with working days.

For problems that fit the restrictions of the linear programming model MAXIMIZE or MINIMIZE use the simplex method to solve the general LP (Linear Programming) problem.

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