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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