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Modeling Summary Types of Constraints Resource Limitations Minimum Requirements Supply and Demand Balance Ratio Control Bounds Accounting Relations Deviation Constraints Approximation or Convexity Constraints Modeling Summary Types of Variables Production Variables Sales Variables Purchase Variables Transformation Variables Slack Variables Surplus Variables Artificial Variables Step Variables Deviation Variables Accounting Variables Modeling Summary Violation of Assumptions Proportionality Max 10Y s.t. Y 6X1 Z1 8Z 2 1.2X 2 3X1 3X 2 4X1 4X 2 0 0 Z1 Z2 4 X1 4 X2 Y, X1 , 0 X2 , Z1 , Z2 , 0 Are the algorithmic assumptions violated? Yes and no. They are not mathematically violated but they are conceptually violated - Decreasing Returns to Scale Modeling Summary Violation of Assumptions Non-Additive Max 3Y s.t. Y Y, 2X1 2X 2 2X 3 0 4X1 2X 2 r1 X1 2X 2 4X 3 r2 X1 , X2 , X3 0. X3 We get more out of using the inputs together, f(X + Y), than we do separately, f(X) + f(Y). The additivity assumption always holds for the individual variables. It may not hold for the model through the combination of variables. Modeling Summary Violation of Assumptions Certainty Max 3Y .3(5X 1 ) .7(4X 2 ) s.t. Y X1 0 X1 2 X2 0 X2 3 X2 0. Y Y, X1 , Are the returns to X certain? Modeling Summary Violation of Assumptions Continuous Max 3X1 2X 2 s.t. X1 X2 10,000 2X1 X2 4,000 Y2 X1 , 0 and integer The solution without the requirement that X1 and X2 be integer is X1 = 4,666 2/3 and X2 = 5,333 1/3. Since X is so large can we round?