Exam 5: Linear Programming: Sensitivity Analysis, Duality, and Specialized Models

In a two-variable graphical linear program, if the RHS of one of the constraints is increased (keeping all other things fixed), the plot of the line will move away from the origin.

When considering simultaneous changes to the objective function coefficients of a linear program, if the sum of the absolute \% changes in the objective function coefficients is less than or equal to $100 \%$ , then the current solution will continue to be the optimal solution, though the value of the objective function may change.

The value of 0 will always be included in any range produced by sensitivity analysis.

When considering simultaneous changes to the RHS of two or more constraints, we add up the \% change of each constraint without ignoring the sign. If the total is less than or equal to 100 , then the current variables will continue to be basic variables in the optimal solution.

Given the following linear programming problem with two non-negative variables $\left(X_{1}\right.$ and $\left.X_{2}\right), 3$ constraints (all $\leq$ type), and a maximize objective function $\left(Y_{i}\right.$ , where $i=1,2,3$ , respective to the dual variables associated with constraints 1,2 and 3), Max: $250 X_{1}+500 X_{2}$ Constraints: $X_{1} \leq 320 \quad 2 X_{1}+5 X_{2} \leq 1100 \quad 1 X_{1}+1.2 X_{2} \leq 480$ Variables are non-negative one of the constraints of the dual problem is

Sensitivity analysis could be used to estimate the range of values of the coefficient of the constraints, taken two at a time, that would keep the same variables as the current solution in the optimal solution (same variables basic) --though the magnitude of the variables may change.

Given the following linear programming problem with two non-negative variables $\left(X_{1}\right.$ and $\left.X_{2}\right)$ and given the range of feasibility for the RHS and range of values for the objective function coefficients that will leave the current solution optimal, answer the question that follows Max: $50 X_{1}+75 X_{2}$ Constraints: $2 X_{1}+5 X_{2} \leq 99 \quad 5 X_{1}+4 X_{2} \leq 120$ Variables are non-negative Range of values of the right hand side values analyzed below: $48 \leq$ RHS $1 \leq 150$ $79.2 \leq$ RHS $2 \leq 247.5$ Suppose that the RHS1 is changed to 82 and RHS2 is changed to 140 simultaneously. Will the current solution be feasible? (same variables remain as basic variables)

Find the dual of the following problem: PRIMAL: Max: $2 X_{1}+4 X_{2}+5 X_{3}$ Subject to: 2+5+3\leq3.02+4+1\leq4.5 \geq0 \geq0 \geq0

In a two-variable graphical linear program, if the RHS of one of the constraints is changed (keeping all other things fixed), then the plot of the corresponding constraint will move in parallel to its old plot.

Every linear programming problem can have two forms, the original formulation (primal) and another form called dual.

Shadow price of resources corresponding to a binding constraint will not change even if the RHS corresponding to one of the binding constraints is changed, as long as the same constraints continue to be binding constraints.

Shadow price of a resource corresponding to a binding constraint may sometimes be 0 .

Sensitivity analysis answers "what if" questions to help the decision maker.

Given the following linear programming problem with two non-negative variables $\left(X_{1}\right.$ and $\left.X_{2}\right)$ , find the range of feasibility for the RHS of constraint 2 (hint: both constraints are binding) Max: $100 X_{1}+200 X_{2}$ Constraints: $2 X_{1}+5 X_{2} \leq 104 \quad 15 X_{1}+3 X_{2} \leq 90$ Variables are non-negative

The dual formulation of a linear programming problem has a maximize objective function, all $\leq$ constraints and non-negative variables, and a minimize objective function, all $\geq$ constraints and nonnegative decision variables.

Shadow price of a resource corresponding to a binding constraint may be positive.

Given the following linear programming problem with two non-negative variables $\left(X_{1}\right.$ and $\left.X_{2}\right), 3$ constraints (all $\leq$ type), and a maximize objective function $\left(Y_{i}\right.$ , where $i=1,2,3$ , respective to the dual variables associated with constraints 1,2 and 3 ), Max: $250 X_{1}+500 X_{2}$ Constraints: $X_{1} \leq 320 \quad 2 X_{1}+5 X_{2} \leq 1100 \quad 1 X_{1}+1.2 X_{2} \leq 480$ Variables are non-negative the variables of the dual problem are required to be

Sensitivity analysis could be used to estimate the range of values of the objective function coefficient, taken two at a time, that would keep the current solution optimal.

In a two-variable linear programming problem, a nonbinding constraint cannot become a binding constraint, even if the RHS of the nonbinding constraint is changed dramatically.

Given the following linear programming problem with two non-negative variables $\left(X_{1}\right.$ and $\left.X_{2}\right)$ and the range of values for the objective function coefficients that will leave the current solution optimal, answer the question that follows Max: $50 X_{1}+75 X_{2}$ Constraints: $2 X_{1}+5 X_{2} \leq 99 \quad 5 X_{1}+4 X_{2} \leq 120$ Variables are non-negative Range of values of the objective function coefficients: $30 \leq \mathrm{C} 1 \leq 93.75$ $40 \leq \mathrm{C} 2 \leq 125$ Suppose that $\mathrm{C} 1$ is changed to 60 and $\mathrm{C} 2$ is changed to 50 simultaneously, will the same solution remain optimal?