## Principles of linear optimisation

Let us consider the following problem.

A professor wishes to work for his university an average of no more than 40 hours per week in view of his family commitment. His contract specifies that on average he must devote at least 10 hours per week to teaching and at least 15 to research. How should he allocate his time?

This can be translated into mathematical formulae as follows:

At least 10 hours per week teaching (x1):

at least 15 hours per week research (x2):

no more than 40 hours per week in view of family commitment:

These inequalities can be represented graphically as shown in Figure 3.1. The shaded area is called the solution space or feasible region, because any combination of the decision variables x1 and x2 in this area satisfies inequalities (3.1), (3.2), (3.3), i.e., the contract requirements and the professor's own wish to work no more than 40 hours/week. The right-hand sides in the three inequalities (3.1), (3.2), (3.3) are called constraints, i.e. boundary conditions limiting the solution space.

Within these constraints an optimum can be established depending on what the professor sees as most desirable (Fig. 3.2):

a. If he wishes to spend as much time as possible on research, he will choose the combination: x1 = 10, x2 = 30;

Figure 3.1 Graphical representation of professor's time allocation problem

Figure 3.2 Professor's time allocation problem b. Conversely, if he likes teaching most, he will choose the combination: x\ = 25, x2 = 15;

c. If he wishes to work as little as possible: x\ = 10, x2 = 15;

d. If he wishes to work as much as possible, dividing his time according to the ratio of his contract: x\ = 16, x2 = 24;

e. If he wishes to work as much as possible, dividing his time equally between teaching and research: x1 = 20, x2 = 20.

The method of establishing an optimum for given constraints is called linear programming, because the constraints are given by linear equations. As we have seen, exactly what is to be optimised depends on personal preference. Only one aspect can be optimised, but the constraints can be many.

Of course, it is quite possible that constraints are defined in such a way that no solution is possible. For instance, if the professor does not wish to work for more than 20 hours - i.e. x\ + x2 < 20 - no solution is possible which satisfies the conditions of his contract. In Open Design, we have to deal with much more than two decision variables and three constraints, in complex cases even several hundreds! The processing of so many variables in an LP model can nowadays be carried out easily by any personal computer.

Efficient, standard software packages are available for this purpose. These packages have been designed for a wide variety of LP problems. As a result, certain features of these software packages are not useful in the specific application of Open Design and can be ignored by the open designer. Conversely, the nature of Open Design problems requires some special 'tricks' when using standard software packages.

To use the LP software package effectively, the open designer has to be familiar with the mathematical model for the general problem of allocating resources to activities. He or she then can 'play' with the program without violating its underlying logic. It is not necessary to have detailed knowledge about how the program finds the optimum using algorithms such as the Simplex Method. It is sufficient to be aware of its essence, namely that it moves in an iterative process, systematically, from one corner-point feasible solution to a better one, until no better corner-point feasible solution can be found. That last corner-point solution is the optimum solution.

For the description of the mathematical model for the general problem of allocating resources to activities, we will use the nomenclature and the standard form adopted in the textbook on Operations Research of Hillier and Lieber-man (2005).

This model is to select the values for the decision variables x1, x2, . . . , xn so as to:

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