Microsoft Excel sensitivity analysis for linear and stochastic program feed formulation

Sensitivity analysis is a part of mathematical programming solutions and is used in making nutritional and economic decisions for a given feed formulation problem. The terms shadow price and reduced cost are familiar linear program (LP) terms to feed formulators. Because of the nonlinear nature of stochastic programming (SP), different methods and terminology are used to define shadow prices and reduced costs. The Lagrange multiplier is used instead of shadow price to describe marginal value of nutrients. Reduced gradient is used instead of reduced cost to describe the price at which ingredients not used in the formulation would be included in the solution. A spreadsheet feed problem was set up with 11 ingredients and 11 constraints. The LP and SP solutions were determined using the Excel Solver algorithm. Two problems compared LP and SP solutions at 50 and 69% probabilities for the protein constraint. All other constraints were held at a 50% probability. Results for the 50% probability comparison showed that the feed formulations, as expected, were the same for both LP and SP. Wheat was not included in the solution. The LP reduced cost and the SP reduced gradient for unused wheat were equivalent. The LP shadow prices and the SP Lagrange multipliers were equivalent. Results for the 69% probability problem showed a difference in the formulated rations. The LP reduced cost was $34.25 and the SP reduced cost was $34.52, showing the respective amounts that the cost of wheat would have to be reduced to be included in the solution. The shadow price and the Lagrange multiplier were $2.73 and $2.71, respectively, for the amount of increase in diet cost that could be expected by a unit of change in the protein requirement. Some differences in precision were noted with the results. A caveat is that, because of nonlinearity, sensitivity analysis for SP is valid only for the single point of the optimal solution.