Question 81: Three Sections Of The Computer Output For A Lin
Question 81three Sections Of The Computer Output For A Linear Program
Three sections of the computer output for a linear program are shown below. Constraint Slack/Surplus Dual Value 2 100.0 0.0 Variable Lower Limit Current Value Upper Limit 1 100.0 130.0 150.0 Constraint Lower Limit Current Value Upper Limit 2 200.0 300.0 400.0 Under which of the following change will the set of binding constraints to the original model be different from the one to the new model? The objective function coefficient for variable 1 increases by 25. The objective function coefficient for variable 1 decreases by 25. The right-hand side value for constraint 2 increases by 50. The right-hand side value for constraint 2 decreases by 50.
Question 9 1. Three sections of the computer output for a linear program are shown below. Constraint Slack/Surplus Dual Value 2 100.0 0.0 Variable Lower Limit Current Value Upper Limit 1 100.0 130.0 150.0 Constraint Lower Limit Current Value Upper Limit 2 200.0 300.0 400.0 What will happen if the objective function coefficient for variable 1 increases by 10? Answer Nothing. The optimal solution and the objective function value at the optimal solution will not change. The objective function value at the optimal solution will change, but the optimal solution will remain the same. The new model will need to be resolved to determine the optimal solution and dual values. None of the above is true.
Question 10 1. Three sections of the computer output for a linear program are shown below. Constraint Slack/Surplus Dual Value 2 100.0 0.0 Variable Lower Limit Current Value Upper Limit 1 100.0 130.0 150.0 Constraint Lower Limit Current Value Upper Limit 2 200.0 300.0 400.0 What will happen if the right-hand side value for constraint 2 increases by 150? Nothing. The optimal solution and the objective function value at the optimal solution will not change. The objective function value at the optimal solution will change, but the optimal solution will remain the same. The new model will need to be resolved to determine the optimal solution and dual values. None of the above is true.
Paper For Above instruction
The analysis of linear programming (LP) output is fundamental to understanding how modifications to the model influence optimal solutions. The three questions provided involve assessing the impact of changes to objective functions and constraints based on LP output, notably focusing on the concept of binding constraints, dual values, and solution stability.
Understanding the LP Output Components
Linear programming outputs typically include information about constraints and decision variables, particularly focusing on slack or surplus values, dual (shadow) prices, variable limits, and current values. Slack or surplus indicates how much a constraint is underutilized or exceeded at the optimal solution. Dual values suggest the marginal worth of relaxing or tightening constraints, providing critical insights into the sensitivity of the solution.
Impact of Changes in Objective Function Coefficients
Question 8 probes the effect of altering the objective function coefficient for a variable. When this coefficient increases by a certain amount, it shifts the optimization landscape, which can potentially change the set of binding constraints—those constraints active at the optimal solution. If the coefficient increases significantly (e.g., by 25), it might make the variable more attractive, thus altering which constraints are binding or active at the optimum. Conversely, decreasing the coefficient might reduce the variable's attractiveness, also influencing binding constraints.
However, small modifications, like an increase of 10, might not sufficiently influence the optimal solution if the variable's current contribution remains less competitive compared to others—particularly if the dual value is zero and the variable is not currently active or binding. The stability depends on the specific LP structure, as indicated by the slack and dual variables.
Changes in RHS Values and Their Effects
Questions 9 and 10 analyze the impact of modifying the right-hand side (RHS) values of constraints. Increasing the RHS value for a constraint generally relaxes it—making it easier for the solution to satisfy—potentially shifting some constraints from active to inactive and thus changing the set of binding constraints. Conversely, decreasing RHS tightens the constraints, possibly activating new constraints or making current ones binding.
In Question 10, increasing the RHS by 150 for constraint 2 is substantial, likely resulting in an altered feasible region, which can change the optimal solution and the objective function value. This necessitates resolving the LP, especially if the previous solution was tight on that constraint.
In contrast, small increases might leave the original solution unchanged if the slack remains significant and the optimal conditions are unaffected, as noted in Question 10's options.
Sensitivity and Stability of LP Solutions
Understanding the sensitivity of the LP solution to changes in the model’s parameters is crucial in practical decision-making. Dual values indicate the rate of change of the objective function concerning RHS variations. Zero dual values often signal that a constraint is non-binding at the current solution, implying that small RHS modifications are unlikely to impact the solution significantly.
Furthermore, the stability of the solution depends on how close the current solution is to the boundary of feasibility—the tighter the constraints, the more sensitive the solution tends to be.
Concluding Insights
These questions highlight essential LP sensitivity analysis principles: changes in objective function coefficients and RHS values can influence the binding constraints and the optimal solution. Determining whether a model's optimal solution would alter involves analyzing slack and dual values, the magnitude of changes, and the current solution's status. As sensitivity analysis reveals, while some changes are inconsequential, others require resolving the LP to accurately identify new optimal conditions.
Hence, practitioners should always evaluate dual values and slack to assess the robustness or sensitivity of the LP solution and ensure informed decision-making under varying scenarios.
References
- Operations Research, 64(4), 927-944.