Linear Programming with Negotiable Constraints

To use linear programming (LP) software effectively, the Open Designer has to be familiar with the mathematical model for the general problem of allocating resources to activities. He or she can then 'play' with the program without violating its underlying logic. It is not necessary to have detailed knowledge about how the program uses the Simplex Method (or faster procedures that are nowadays available) to find the optimum. 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 general mathematical model, we will use the nomenclature and the standard form adopted in the Operations Research textbook of Hillier and Lieberman (2005). This model is used to select the values for the decision variables x1, x2, . . . , xn so as to:

subject to the restrictions:

am1 x1 + am2x2 + ' ' ' + amnxn < bm and x1 > 0, x2 > 0 xn > 0

For the sake of brevity, we use £ notation and write:

subject to:

Figure 1.1 What's Best! toolbar and

This is adopted as the standard form for the linear programming problem. Any situation whose mathematical formulation fits this model is a linear programming model.

The function Z being maximised, c1 x1 + c2x2 + ■ ■ ■ + cnxn, is called the objective function. The decision variables - the Xj - are sometimes referred to as the uncontrolled or endogenous variables. The input variables - the aj, b, and cj -may be referred to as parameters of the model or as the controlled or exogenous variables. The restrictions are referred to as constraints. The first m constraints, b1, b2,..., bm (those with a function ai1 x1 + ai2x2 + ■ ■ ■ + ainxn representing the total usage of resource i, on the left) are called functional constraints. The xj > 0 restrictions are called non-negativity constraints.

In traditional linear programming, the constraints bi, b2, b?,..., b m are considered to be fixed. Often they represent physical constraints that indeed cannot be changed, such as the dimensions (b1, b2, . . . , bm) of land available to grow various vegetables (x1, x2,..., xn). In Open Design, by contrast, at least some of the constraints b1, b2, . . . , bm are considered to be negotiable. This is a fundamental difference with traditional linear programming, which has far reaching consequences in practice. It means that the mathematical outcome infeasible can be changed into feasible after all.

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