this case, all the unwanted deviations are multiplied by weights, reflecting their relative importance, and added together as a single sum to form the achievement function. Deviations measured in different units cannot be summed directly due to the phenomenon of incommensurability. Hence each unwanted deviation is multiplied by a normalisation constant to allow direct comparison. Popular choices for normalisation constants are the goal target value of the corresponding objective (hence turning all deviations into percentages) or the range of the corresponding objective (between the best and the worst possible values, hence mapping all deviations onto a zero-one range). For decision makers more interested in obtaining a balance between the competing objectives, Chebyshev goal programming is used. Introduced by Flavell in 1976, this variant seeks to minimise the maximum unwanted deviation, rather than the sum of deviations. This utilises the Chebyshev distance metric. Strengths and weaknesses A major strength of goal programming is its simplicity and ease of use. This accounts for the large number of goal programming applications in many and diverse fields. Linear goal programmes can be solved using linear programming software as either a single linear programme, or in the case of the lexicographic variant, a series of connected linear programmes. Goal programming can hence handle relatively large numbers of variables, constraints and objectives. A debated weakness is the ability of goal programming to produce solutions that are not Pareto efficient. This violates a fundamental concept of decision theory, that no rational decision maker will knowingly choose a solution that is not Pareto efficient. However, techniques are available to detect when this occurs and project the solution onto the Pareto efficient solution in an appropriate manner. The setting of appropriate weights in the goal programming model is another area that has caused debate, with some authors suggesting the use of the analytic hierarchy process or interactive methods for this purpose. Also, the weights of the objective functions can be calculated based on their preference using the ordinal priority approach. See also Decision-making software External links LiPS — Free easy-to-use GUI program intended for solving linear, integer and goal programming problems. LINSOLVE - Free Windows command-line window linear programming and linear goal programming] References Mathematical optimization Multiple-criteria decision analysis Goal de:Entscheidung unter Sicherheit#Zielprogrammierung