Find the dual of the following problem: PRIMAL: Max: $2 X_{1}+4 X_{2}+5 X_{3}$ Subject to: \[\begin{array}{rrr} 2 X_{1}+5 X_{2}+3 X_{3} \geq 3.0 \quad 2 X_{1}+4 X_{2}+1 X_{3} \leq 50 \quad 3 X_{1}+4 X_{2}+2 X_{3}=20 & X_{1} \geq 0 \\ & X_{2} \geq 0 \\ & X_{3} \geq 0 \end{array}\]

First of all, the Primal problem must be

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

Wilkinson Auto Dealership sells standard automobiles and station wagons. The profit contribution for automobiles is $\$ 250.00$ per unit and $\$ 500.00$ per unit for station wagons. The company is planning the placement of orders with the manufacturer for next quarter. Orders for automobiles and station wagons can not exceed 320 and 160, respectively. Dealer preparation takes 2 hrs/auto and 5.0 hrs/wagon. They have 1100 hrs of preparation time next quarter. Autos take 1 unit of space, whereas wagons take 1.2 units of space. 480 units of space are available. In order to maintain some balance, the number of cars ordered should not be more than $150 \%$ the number of wagons ordered. Assume that they can sell all the autos and wagons they order for the quarter. The formulation of the problem is given below. Using excel-solver, completea sensitivity analysis of the objective function coefficients and report on the range of values of the coefficients that will leave the current solution optimal find the range of feasibility for changes in the RHS of each constraint Decision variables: Let $X_{1}$ be the number of automobiles and $X_{2}$ be the number of station wagons ordered next quarter. Objective function: Max: $250 X_{1}+500 X_{2}$ Constraints: $X_{1} \leq 320 \quad X_{2} \leq 160 \quad 2 X_{1}+5 X_{2} \leq 1100 \quad 1 X_{1}+1.2 X_{2} \leq 480 \quad X_{1} \leq 1.5 X_{2}$ Variables are non-negative Optimal solution using Solver: $X_{1}=206.25 \quad X_{2}=137.5$ The objective function value corresponding to the optimal solution is: $\$ 120.312 .50$

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Question 1

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(A) Output from sensitivity analysis is given below:
$(A) Output from sensitivity analysis is given below: Using this data, the range on the coefficient of X_{1} is: 200 \leq \mathrm{C} 1<\infty Similarly, range on the coefficient of \mathrm{X}_{2} is: -375 \leq \mathrm{C} 2 \leq 625 (B) Output for the ranges of feasibility of RHS from solver is given below: 206.25 \leq RHS 1<\infty 137.50 \leq RHS 2<\infty 0 \leq RHS3 \leq 1280 371.25 \leq RHS 4<\infty -90.00 \leq RHS5 \leq 182$ Using this data, the range on the coefficient of $X_{1}$ is: $200 \leq \mathrm{C} 1<\infty$
Similarly, range on the coefficient of $\mathrm{X}_{2}$ is: $-375 \leq \mathrm{C} 2 \leq 625$
(B) Output for the ranges of feasibility of RHS from solver is given below:
$(A) Output from sensitivity analysis is given below: Using this data, the range on the coefficient of X_{1} is: 200 \leq \mathrm{C} 1<\infty Similarly, range on the coefficient of \mathrm{X}_{2} is: -375 \leq \mathrm{C} 2 \leq 625 (B) Output for the ranges of feasibility of RHS from solver is given below: 206.25 \leq RHS 1<\infty 137.50 \leq RHS 2<\infty 0 \leq RHS3 \leq 1280 371.25 \leq RHS 4<\infty -90.00 \leq RHS5 \leq 182$ $206.25 \leq$ RHS $1<\infty$
$137.50 \leq$ RHS $2<\infty$
$0 \leq$ RHS3 $\leq 1280$
$371.25 \leq$ RHS $4<\infty$
$-90.00 \leq$ RHS5 $\leq 182$

In a two-variable graphical linear program, if the RHS of one of the constraints is changed (keeping all other things fixed), then the problem cannot become infeasible.

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Question 2

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False

The value of ${ }^{\infty}$ may be included as part of the range produced by sensitivity analysis in some problems.

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Question 3

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True

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 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

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Question 4

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

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Question 5

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 $X_{2}$ 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

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Question 6

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 $X_{2}$ 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

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Question 7

In order to consider changes to the objective function coefficients and the RHS simultaneously, one can apply the $100 \%$ rule - total \% changes must be less than 100 - where negative change is also added without the negative sign.

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Question 8

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 objective function of the dual problem is

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Question 9

Sensitivity analysis could be used to estimate the range of values of the right hand side, taken one 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.

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Question 10

Find the dual of the following problem: PRIMAL: Max: $2 X_{1}+4 X_{2}+5 X_{3}$ Subject to: 2+5+3\geq3.02+4+1\leq503+4+2=20 \geq0 \geq0 \geq0

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Question 11

The lower bound of the feasibility range of a nonbinding constraint is determined by decreasing the RHS of the constraint by the amount of surplus.

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Question 12

In a two-variable linear programming problem, if the RHS corresponding to a binding constraint were to be increased, the value of the variables corresponding to the optimal solution would also change.

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Question 13

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 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

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Question 14

Since the solution to the primal problem also contains the solution to the dual problem, there is no need to study dual problems.

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Question 15

If a problem has a $\leq$ constraint with a positive RHS, and if that resource is not fully utilized in the optimal solution, then the upper limit on the range using sensitivity analysis for that RHS will be $\infty$ .

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Question 16

If the primal problem has maximize objective function, non-negative variables, and 4 constraints, each of which is $\leq$ type, then the dual problem will have 4 variables and all of them will be non-negative.

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Question 17

Any linear programming problem can be rewritten as an equivalent linear programming problem with maximize objective function, all $\leq$ constraints and non-negative variables.

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Question 18

Sensitivity analysis could be used to estimate the range of values of the coefficient of the constraints, taken one 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.

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Question 19

The value of ${ }^{\infty}$ will always be included in any range produced by sensitivity analysis.

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Question 20

Showing 1 - 20 of 55

In a two-variable graphical linear program, if the RHS of one of the constraints is changed (keeping all other things fixed), then the problem cannot become infeasible.

The value of ${ }^{\infty}$ may be included as part of the range produced by sensitivity analysis in some problems.

In order to consider changes to the objective function coefficients and the RHS simultaneously, one can apply the $100 \%$ rule - total \% changes must be less than 100 - where negative change is also added without the negative sign.

Sensitivity analysis could be used to estimate the range of values of the right hand side, taken one 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.

Find the dual of the following problem: PRIMAL: Max: $2 X_{1}+4 X_{2}+5 X_{3}$ Subject to: 2+5+3\geq3.02+4+1\leq503+4+2=20 \geq0 \geq0 \geq0

The lower bound of the feasibility range of a nonbinding constraint is determined by decreasing the RHS of the constraint by the amount of surplus.

In a two-variable linear programming problem, if the RHS corresponding to a binding constraint were to be increased, the value of the variables corresponding to the optimal solution would also change.

Since the solution to the primal problem also contains the solution to the dual problem, there is no need to study dual problems.

If a problem has a $\leq$ constraint with a positive RHS, and if that resource is not 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, non-negative variables, and 4 constraints, each of which is $\leq$ type, then the dual problem will have 4 variables and all of them will be non-negative.

Any linear programming problem can be rewritten as an equivalent linear programming problem with maximize objective function, all $\leq$ constraints and non-negative variables.

Sensitivity analysis could be used to estimate the range of values of the coefficient of the constraints, taken one 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.

The value of ${ }^{\infty}$ will always be included in any range produced by sensitivity analysis.

Introduction to Management Science, Modeling, and Excel Spreadsheets

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Linear Programming: Basic Concepts and Graphical Solutions

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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 changed (keeping all other things fixed), then the problem cannot become infeasible.

The value of ∞{ }^{\infty}∞ may be included as part of the range produced by sensitivity analysis in some problems.

In order to consider changes to the objective function coefficients and the RHS simultaneously, one can apply the 100%100 \%100% rule - total \% changes must be less than 100 - where negative change is also added without the negative sign.

Sensitivity analysis could be used to estimate the range of values of the right hand side, taken one 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.

Find the dual of the following problem: PRIMAL: Max: 2X1+4X2+5X32 X_{1}+4 X_{2}+5 X_{3}2X1​+4X2​+5X3​ Subject to: 2+5+3\geq3.02+4+1\leq503+4+2=20 \geq0 \geq0 \geq0

The lower bound of the feasibility range of a nonbinding constraint is determined by decreasing the RHS of the constraint by the amount of surplus.

In a two-variable linear programming problem, if the RHS corresponding to a binding constraint were to be increased, the value of the variables corresponding to the optimal solution would also change.

Since the solution to the primal problem also contains the solution to the dual problem, there is no need to study dual problems.

If a problem has a ≤\leq≤ constraint with a positive RHS, and if that resource is not 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, non-negative variables, and 4 constraints, each of which is ≤\leq≤ type, then the dual problem will have 4 variables and all of them will be non-negative.

Any linear programming problem can be rewritten as an equivalent linear programming problem with maximize objective function, all ≤\leq≤ constraints and non-negative variables.

Sensitivity analysis could be used to estimate the range of values of the coefficient of the constraints, taken one 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.

The value of ∞{ }^{\infty}∞ will always be included in any range produced by sensitivity analysis.

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

The value of ${ }^{\infty}$ may be included as part of the range produced by sensitivity analysis in some problems.

In order to consider changes to the objective function coefficients and the RHS simultaneously, one can apply the $100 \%$ rule - total \% changes must be less than 100 - where negative change is also added without the negative sign.

Find the dual of the following problem: PRIMAL: Max: $2 X_{1}+4 X_{2}+5 X_{3}$ Subject to: 2+5+3\geq3.02+4+1\leq503+4+2=20 \geq0 \geq0 \geq0

If a problem has a $\leq$ constraint with a positive RHS, and if that resource is not 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, non-negative variables, and 4 constraints, each of which is $\leq$ type, then the dual problem will have 4 variables and all of them will be non-negative.

Any linear programming problem can be rewritten as an equivalent linear programming problem with maximize objective function, all $\leq$ constraints and non-negative variables.

The value of ${ }^{\infty}$ will always be included in any range produced by sensitivity analysis.