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

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 and range of values for the objective function coefficients that will leave the current solution optimal. You may use graphical method for your analysis, since the problem has only 2 decision variables 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

(Essay)

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Question 41

Shadow price of a resource corresponding to a nonbinding constraint is always 0 .

(True/False)

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Question 42

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 more than $100 \%$ , then the current solution will not be optimal.

(True/False)

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Question 43

If a problem has a $\leq$ constraint with a positive RHS, and if that resource is fully utilized in the optimal solution, then the upper limit on the range using sensitivity analysis for that RHS will be ${ }^{\infty}$ .

(True/False)

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Question 44

If a problem has a $\leq$ constraint with a positive RHS, then decreasing the RHS cannot improve (i.e. increase the objective function value corresponding to the optimal solution if it is a "maximize" objective function and decrease it if it is a "minimize" objective function) the objective function value.

(True/False)

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Question 45

In a two-variable graphical linear program, if the coefficient of one of the variables in the objective function is changed (while the other remains fixed), then the slope of the objective function expression will change.

(True/False)

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Question 46

In a two-variable linear programming problem, even if the RHS corresponding to a binding constraint were to be increased by a very large amount, it will always continue to be a binding constraint corresponding to the changed problem.

(True/False)

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Question 47

If a problem has a $\leq$ constraint with a positive RHS, then increasing the RHS would leave the optimal solution's objective function value the same or improve it (i.e. increase the objective function value corresponding to the optimal solution if it is a "maximize" objective function and decrease it if it is a "minimize" objective function).

(True/False)

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Question 48

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

(True/False)

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Question 49

Shadow price of a resource corresponding to a binding constraint in a problem with maximize profit as its objective function

(Multiple Choice)

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Question 50

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 1 (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

(Multiple Choice)

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Question 51

Given the following linear programming problem with two non-negative variables $\left(X_{1}\right.$ and $\left.X_{2}\right)$ , find the range of values for the objective function coefficient of $X_{1}$ that will leave the current solution optimal (that is range of optimality or range of insignificance as the case may be) (hint: both constraints are not binding) Max: $10 X_{1}+200 X_{2}$ Constraints: $2 X_{1}+5 X_{2} \leq 100 \quad 15 X_{1}+3 X_{2} \leq 90$ Variables are non-negative

(Multiple Choice)

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Question 52

Given the following linear programming problem with two non-negative variables $\left(X_{1}\right.$ and $X_{2}$ ), find the range of values for the objective function coefficient of $X_{1}$ that will leave the current solution optimal (that is range of optimality or range of insignificance as the case may be) (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

(Multiple Choice)

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Question 53

If the primal problem has maximize objective function with $\leq$ constraints and non-negative variables, the dual will have minimize objective function with $\geq$ constraints and strictly negative variables.

(True/False)

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Question 54

In a linear program, even if RHS of constraint/s is/are changed with the range of feasibility, the shadow price of one or more resources may change.

(True/False)

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Question 55

Shadow price of a resource corresponding to a nonbinding constraint is always 0 .

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 more than $100 \%$ , then the current solution will not be optimal.

If a problem has a $\leq$ constraint with a positive RHS, and if that resource is fully utilized in the optimal solution, then the upper limit on the range using sensitivity analysis for that RHS will be ${ }^{\infty}$ .

In a two-variable graphical linear program, if the coefficient of one of the variables in the objective function is changed (while the other remains fixed), then the slope of the objective function expression will change.

In a two-variable linear programming problem, even if the RHS corresponding to a binding constraint were to be increased by a very large amount, it will always continue to be a binding constraint corresponding to the changed problem.

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

Shadow price of a resource corresponding to a binding constraint in a problem with maximize profit as its objective function

If the primal problem has maximize objective function with $\leq$ constraints and non-negative variables, the dual will have minimize objective function with $\geq$ constraints and strictly negative variables.

In a linear program, even if RHS of constraint/s is/are changed with the range of feasibility, the shadow price of one or more resources may change.

Introduction to Management Science, Modeling, and Excel Spreadsheets

Forecasting

Linear Programming: Basic Concepts and Graphical Solutions

Linear Programming: Applications and Solutions

Transportation, Assignment, and Transshipment Problems

Integer Programming

Network Optimization Models

Nonlinear Optimization Models

Multi-Criteria Models

Decision Theory

Markov Analysis

Waiting Line Models

Simulation Cdrom Modules

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

Shadow price of a resource corresponding to a nonbinding constraint is always 0 .

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 more than 100%100 \%100% , then the current solution will not be optimal.

If a problem has a ≤\leq≤ constraint with a positive RHS, and if that resource is fully utilized in the optimal solution, then the upper limit on the range using sensitivity analysis for that RHS will be ∞{ }^{\infty}∞ .

In a two-variable graphical linear program, if the coefficient of one of the variables in the objective function is changed (while the other remains fixed), then the slope of the objective function expression will change.

In a two-variable linear programming problem, even if the RHS corresponding to a binding constraint were to be increased by a very large amount, it will always continue to be a binding constraint corresponding to the changed problem.

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

Shadow price of a resource corresponding to a binding constraint in a problem with maximize profit as its objective function

If the primal problem has maximize objective function with ≤\leq≤ constraints and non-negative variables, the dual will have minimize objective function with ≥\geq≥ constraints and strictly negative variables.

In a linear program, even if RHS of constraint/s is/are changed with the range of feasibility, the shadow price of one or more resources may change.

Introduction to Management Science, Modeling, and Excel Spreadsheets

Forecasting

Linear Programming: Basic Concepts and Graphical Solutions

Linear Programming: Applications and Solutions

Transportation, Assignment, and Transshipment Problems

Integer Programming

Network Optimization Models

Nonlinear Optimization Models

Multi-Criteria Models

Decision Theory

Markov Analysis

Waiting Line Models

Simulation Cdrom Modules

Filters

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 more than $100 \%$ , then the current solution will not be optimal.

If a problem has a $\leq$ constraint with a positive RHS, and if that resource is fully utilized in the optimal solution, then the upper limit on the range using sensitivity analysis for that RHS will be ${ }^{\infty}$ .

If the primal problem has maximize objective function with $\leq$ constraints and non-negative variables, the dual will have minimize objective function with $\geq$ constraints and strictly negative variables.