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OR-Notes are a series of introductory notes on topics that fall under the broad heading of the field of operations research (OR). They were originally used by me in an introductory OR course I give at Imperial College. They are now available for use by any students and teachers interested in OR subject to the following Cleo sandals White Senso Hk6wAyLx
.

A full list of the topics available in OR-Notes can be found .

You will recall from the FOOTWEAR Ankle boots on YOOXCOM Giancarlo Paoli m84mR
that the conditions for a mathematical model to be a linear program (LP) were:

LP's are important - this is because:

We will return later to the simplex algorithm for panelled sneakers White Osklen f6UWf
but for the moment we will concentrate upon formulating LP's.

Some of the major application areas to which LP can be applied are:

We consider below some specific examples of the types of problem that can be formulated as LP's. Note here that the key to formulating LP's is practice . However a useful hint is that common objectives for LP's are minimise cost/maximise profit .

A bank makes four kinds of loans to its personal customers and these loans yield the following annual interest rates to the bank:

The bank has a maximum foreseeable lending capability of £250 million and is further constrained by the policies:

Formulate the bank's loan problem as an LP so as to maximise interest income whilst satisfying the policy limitations.

Note here that these policy conditions, whilst potentially limiting the profit that the bank can make, also limit its exposure to risk in a particular area. It is a fundamental principle of risk reduction that risk is reduced by spreading money (appropriately) across different areas.

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- namely

Note here that as in all formulation exercises we are translating a verbal description of the problem into an equivalent mathematical description.

A useful tip when formulating LP's is to express the variables, constraints and objective in words before attempting to express them in mathematics.

Essentially we are interested in the amount (in £) the bank has loaned to customers in each of the four different areas (not in the actual number of such loans). Hence let

x i = amount loaned in area i in £m (where i=1 corresponds to first mortgages, i=2 to second mortgages etc)

and note that x i >= 0 (i=1,2,3,4).

Note here that it is conventional in LP's to have all variables >= 0. Any variable (X, say) which can be positive or negative can be written as X 1 -X 2 (the difference of two new variables) where X 1 >= 0 and X 2 >= 0.

(a) limit on amount lent

x 1 + x 2 + x 3 + x 4 <= 250

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