© 2000 John Petroff
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© 2000 John Petroff |
G- Linear programming and "what if" iterations
Linear programming is used in financial analysis to find solutions in cases of constrained optimization. A typical example of where this is necessary is in capital budgeting, as illustrated in Chapter 10 Section E-7. A good web site to look for information, software and resources related to linear programming is the Linear Programming Frequently Asked Qeustions of Northwestern University and Argone National Laboratory at http://www-unix.mcs.anl.gov/otc/Guide/faq/linear-programming-faq.html. A detailed step by step introduction to linear programming is attributable to Sturm.
Just as for data smoothing in the previous section, this is a procedure most appropriate for inside analysts. The goal of the procedure is to optimize (i.e. maximum or minimum) some variable Z (such as profit, wealth, sales, unit costs, expenses, cash flow or cash position) given by an objective function subject to a set of n constraints expressed as a system of inequations. For instance, one could have an objective function such as
Z = a1x1 + a2x2 + . . . + akxk
and a set of constraints
Naturally the inequations can be formulated as greater-than-or-equal or smaller-than-or-equal, and not all the controllable decision variables need be in each functional relationship. The procedure requires that the controllable decision variables be first identified, then the objective function stated as a linear equation, and finally that the constraints (e.g. limited scarce resources such as amount of capital available, size of plant, number of employees) be stated as linear functions of the controllable decision variables. If there are more than one objective function, the procedure is called goal programming.
The coefficients of the function must be obtained from financial or production information, or by running regressions. The inequations are converted into equations by incorporating in each a slack variable (which can only be either negative or positive). The solution is found by a simplex algorithm, i.e. placing all variables in a tableau (i.e. spreadsheet), and values are entered and modified in a series of iterations of feasible solutions that satisfy all equations, until additional iterations no longer permit feasible solutions.
-- example --
The procedure allows more than a determination of just the largest profit or lowest cost, but also what amounts of various inputs are necessary to achieve the desired goal. In addition, the method can also show which of the constraints is the most damaging, and suggests, therefore, a desirable management strategy. Firms are known to apply the technique in capital budgeting, short-term and long-term planning, and other decision analysis. The method has also been used for analysis of portfolios.
A modern version of linear programming is present in many software packages and spreadsheets, that tests various assumptions to achieve a stated goal. In some of these packages, it is referred to as the "what if" procedure which goes through iterations on a given set of financial statements and stated relationships. The procedures are also sometime called sensitivity analysis (not to be confused with the sensitivity analysis discussed previously as a form of regression analysis) and simulation (which will be touched upon again in the following section).
See review questions Q-5G.1 through Q-5G.3.
See research assignment R-5.11.
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