Given this linear programming model, solve the model and then answer the questions

a
Here is a tip:
Constraints are the restrictions or the limitations of the decision variables.

Explanation
The machine constraints and materials seem to be binding with:

Machine: Decision variable of 160
Material: Decision variable of 200
Verified Answer
Yes, the machine constraints and materials are binding.

b
Here is a tip:
Objective function is a tool to maximize or minimize something.

Explanation
The range of optimality for the objective function coefficient of product, 3, is from 15-1.5 to 15+21. So, the values of decision variables value would not be changed because of the increase from 15 to 22.
There will be a change in the value of objective function. The change in the objective function is calculated by adding the value of objective function to the product of decision variable (x3) and difference between increased profit and initial profit.
\begin{aligned} \text{Change in ObjectiveFunction}\left( \text{Z} \right)&=\text{Value of Objective Function} \\ & +\text{Decision Variable}\left( {{\text{x}}_{\text{3}}} \right)\times \left[ \text{Increased Profit}-\text{Initial Profit} \right] \\ &=\$792+48\times\left(\$22-\$15\right)\\&=\$1,128\end{aligned}
Change in ObjectiveFunction(Z) ​=Value of Objective Function+Decision Variable(x3​)×[Increased Profit−Initial Profit]=$792+48×($22−$15)=$1,128​
Verified Answer
There will be no change in the values of decision variables.
The objective function value will increase to Z = $1,128

c
Here is a tip:
Decision variables are quantities that need to be determined in order to solve a problem.

Explanation
The range of optimality for the objective function coefficient of Product 1 is from infinity to 22.2. As 22 is within the range, there is going to be no change in value of the decision variables.

There are no units being produced for Product 1. There is going to be no change in the objective function value as well.

Verified Answer
The decision variable values as well as the values of objective function are not going to change.

d
Here is the step-by-step explanation, verified by an educator:

Step 1 of 3
Calculate the slack in labor constraint by subtracting the left-hand side value of labor constraint from the right-hand side value of labor constraint.

\begin{aligned} \text{Slack in Labor Constraint}&=\text{Right-hand Side Value of Labor Constraint} - \text{Left-hand Side Value of Labor Constraint} \\ &=288-232 \\ &=56 \end{aligned} Slack in Labor Constraint ​=Right-hand Side Value of Labor Constraint−Left-hand Side Value of Labor Constraint=288−232=56 ​

Step 2 of 3
Calculate the available labor hours by subtracting the reduction in labor hours from the right-hand side value of labor constraint.

\begin{aligned} \text{Available Labor Hours} &=\text {Right-hand Side Value of Labor Constraint} -\text {Reduction in Labor Hours} \\ &=288-10 \\ &=278 \end{aligned}
Available Labor Hours ​=Right-hand Side Value of Labor Constraint−Reduction in Labor Hours=288−10=278​
Step 3 of 3
Calculate the new slack in labor constraint by subtracting the reduction in hours from the slack in labor constraint.

\begin{aligned} \text{New Slack in Labor Constraint}&=\text{Slack in Labor Constraint}-\text{Reduction in Labor Hours} \\ &=56-10 \\ &=46 \end{aligned}
New Slack in Labor Constraint ​=Slack in Labor Constraint−Reduction in Labor Hours=56−10=46 ​
Final answer
There is no change in the decision variables value. The slack in labor constraint is 56 hours. The range of feasibility for labor constraint is 232 to 288+1E+30 or 232 to +∞. The reduction in labor hours is not going to affect the decision variables. The available labor hours are 278 and that is still in the range of feasibility.
The shadow price in the sensitivity report is 0. There is going to be no change in objective function. The new slack of labor constraint is 46.

e
Here is a tip:
Additional profit is the increase in profit caused by a change in the constraints for an objective function.

Explanation
The production of 20 units of commodity 2 in place of 16 will not fetch any extra profit as the constraint of commodity 2 is not obligatory. This is due to the fact that the value of the decision variables and the objective function will not change.

Verified Answer
There will not be any additional profit generated due to no change in the value of decision variables and objective function.

f
Here is the step-by-step explanation, verified by an educator:

Step 1 of 7
Calculate the ratio of change in Product 1 by dividing an increase in profit per unit by the allowable increase for Product 1.

\begin{aligned} \text{Ratio of Change in Product 1}&=\frac{\text{Increase in Profit per Unit}}{\text{Allowable Increase for Product 1}} \\ &=\frac{1}{10.2} \\ &=0.098 \end{aligned}
Ratio of Change in Product 1 ​=Allowable Increase for Product 1Increase in Profit per Unit​=10.21​=0.098 ​
Step 2 of 7
Calculate the ratio of change in Product 2 by dividing the increase in profit per unit by the allowable increase for Product 2.

\begin{aligned} \text{Ratio of Change in Product 2}&=\frac{\text{Increase in Profit per Unit}}{\text{Allowable Increase for Product 2}} \\ &=\frac{1}{2} \\ &=0.500 \end{aligned}
Ratio of Change in Product 2 ​=Allowable Increase for Product 2Increase in Profit per Unit​=21​=0.500 ​
Step 3 of 7
Calculate the ratio of change in Product 3 by dividing an increase in profit per unit by the allowable increase for Product 3.

\begin{aligned} \text{Ratio of Change in Product 3}&=\frac{\text{Increase in Profit per Unit}}{\text{Allowable Increase for Product 3}} \\ &=\frac{1}{21} \\ &=0.048 \end{aligned}
Ratio of Change in Product 3 ​=Allowable Increase for Product 3Increase in Profit per Unit​=211​=0.048 ​
Step 4 of 7
Calculate the total of the change in the ratio by adding the ratio of change in Product 1, the ratio of change in Product 2, and the ratio of change in Product 3.

\begin{aligned} \text{Total of the Change in the Ratio}&=\text{Ratio of Change in Product 1}+\text{Ratio of Change in Product 2}+\text{Ratio of Change in Product 3} \\ &=0.098+0.500+0.048 \\ &=0.646 \end{aligned}
Total of the Change in the Ratio ​=Ratio of Change in Product 1+Ratio of Change in Product 2+Ratio of Change in Product 3=0.098+0.500+0.048=0.646 ​
Step 5 of 7
Calculate the new decision variable x2 objective by adding the decision variable x2 objective and the increase in profit.

\begin{aligned} \text{New Decision Variable }{{\text{x}}_{2}}\text{\,Objective}&=\text{Decision Variable }{{\text{x}}_{2}}\text{\,Objective}+\text{Increase in Profit} \\ &=18+1 \\ &=19 \end{aligned}
New Decision Variable x2​Objective ​=Decision Variable x2​Objective+Increase in Profit=18+1=19 ​
Step 6 of 7
Calculate the new decision variable x3 objective by adding the decision variable x3 objective and the increase in profit.

\begin{aligned} \text{New Decision Variable }{{\text{x}}_{3}}\text{\,Objective}&=\text{Decision Variable }{{\text{x}}_{3}}\text{\,Objective}+\text{Increase in Profit} \\ &=15+1 \\ &=16 \end{aligned}
New Decision Variable x3​Objective ​=Decision Variable x3​Objective+Increase in Profit=15+1=16 ​
Step 7 of 7
Calculate the value of the objective function by adding the product of the decision variable x2 and the new decision variable x2 objective and the product of the decision variable x3 and the new decision variable x3 objective.

\begin{aligned} \text{Value of the Objective Function} &=\text{(Decision Variable }{{\text{x}}_{2}}\times \text{New Decision Variable }{{\text{x}}_{\text{2}}}\text{\,Objective)} \\&+\text{(Decision Variable }{{\text{x}}_{3}}\times \text{New Decision Variable }{{\text{x}}_{3}}\text{\,Objective)} \\ &=\text{(4}\times \text{19)+(48}\times \text{16)} \\ &=76+768 \\ &=844 \end{aligned}
Value of the Objective Function ​=(Decision Variable x2​×New Decision Variable x2​Objective)+(Decision Variable x3​×New Decision Variable x3​Objective)=(4×19)+(48×16)=76+768=844 ​
Final answer
The optimal value of the decision variable is not going to change because the value of the ratio of change in products is 0.646, which is within the range of 1.
The objective function value is going to change and its optimal value is $844.