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Shadow Prices, Sensitivity Analysis, and Column Generation
Published in Craig A. Tovey, Linear Optimization and Duality, 2020
If there is a slack variable xk in the constraint Ai,:x=bi, then the reduced cost of xk equals the negative of the dual variable πi for constraint i if the constraint had been a ≤ constraint prior to the introduction of xk into the model. Question: Why the negative?4 If the constraint had been a ≥ constraint, the value is not multiplied by −1 because in that case A:,k=−ei. If you are ever confused about the meaning of a dual variable, remember that relaxing a constraint can only be harmless or beneficial, and tightening a constraint can only be harmless or harmful. If the constraint was originally an equality constraint, use Equation (7.1) to determine the meaning.
Sensitivity Analysis
Published in Timothy R. Anderson, Optimization Modeling Using R, 2023
These results are interesting. The reduced costs of variables that are between simple upper and lower bounds will be zero. Again, the reduced cost of a variable is the difference between the value of the resources consumed by the product and the value of the product in the objective function. All of our product's variables have simple lower bounds of zero and no upper bounds. Ants and Cats have zero reduced cost while Bats have a negative reduced cost. Let's see if this is consistent with the interpretation of reduced costs.
Linear programming for mining systems
Published in Amit Kumar Gorai, Snehamoy Chatterjee, Optimization Techniques and their Applications to Mine Systems, 2023
Amit Kumar Gorai, Snehamoy Chatterjee
Case 2: Changes in the coefficients in the objective function: In the second case, the sensitivity analysis is done to understand the change in the objective function value per unit change in the value of decision variables. The reduced cost for a decision variable is nonzero only when the variable’s value is equal to its upper or lower bound at the optimal solution.
A column generation based approach for an integrated production and transportation scheduling problem with dual delivery modes
Published in International Journal of Production Research, 2023
Kunpeng Li, Peiyang He, P. N. Ram Kumar
The branch-and-price algorithm combines column generation and branch-and-bound algorithm (Desrochers, Desrosiers, and Solomon 1992). The procedure typically begins by forming a master problem and a sub-problem (Dantzig and Wolfe 1960). Instead of explicitly enumerating all the columns (variables), the Restricted Linear Master Problem (RLMP) considers only a subset of columns and is solved by relaxing the integrality restrictions on the variables. A sub-problem called the pricing problem is solved to find columns with a negative reduced cost that can enter the basis and improve the objective function value. Branching happens only when no columns can enter the basis, and the solution to the relaxation does not satisfy the integrality conditions.