Question

In linear programming context, sensitivity analysis is a technique to\[\]
A. Allocate resources optimally\[\]
B. Minimize cost operations \[\]
C. Spell out the relation between primal and dual\[\]
D. Determine how optimal solution in response to problem units\[\]

Answer 1

Hint: We recall that sensitivity analysis is the study of the output with respect to small changes in inputs. We determine what is the output in a linear programming problem and what are the inputs and accordingly choose the correct option. \[\]

Complete step by step answer:
Sensitivity analysis in general terms is the study of uncertainty in the output of any mathematical model or system can be divided and allocated to new inputs with changes in varying degree. It determines how sensitive the output with respect to small changes in inputs. Sensitivity analysis is used in industry to test robustness, understand the relationship between input and output , finding error in the output etcetera.
We know that in linear programming problem or LPP the output is the cost function. The cost function is the function which has to be optimized that is minimized or maximized. It is expressed in the standard from of the LPP with $n$ linear variables ${{x}_{1}},{{x}_{2}},...,{{x}_{n}}$ and their respective costs ${{c}_{1}},{{c}_{2}},...{{c}_{n}}$ as
\[C\left( x \right)={{c}_{1}}{{x}_{1}}+{{c}_{2}}{{x}_{2}}+...{{c}_{n}}\]
We suppose that the cost has to be minimized. We define the problem constraints(also called problem units) for the cost function as
\[\begin{align}
  & {{a}_{11}}{{x}_{1}}+{{a}_{12}}{{x}_{2}}+...+{{a}_{1n}}{{x}_{n}}\le {{b}_{1}} \\
 & {{a}_{11}}{{x}_{1}}+{{a}_{22}}{{x}_{2}}+...+{{a}_{2n}}{{x}_{n}}\le {{b}_{2}} \\
 & \vdots \\
 & {{a}_{m1}}{{x}_{1}}+{{a}_{m2}}{{x}_{2}}+...+{{a}_{mn}}{{x}_{n}}\le {{b}_{m}} \\
\end{align}\]
The non-negative constraint is,
\[{{x}_{1}},{{x}_{2}},...,{{x}_{n}}\ge 0\]
If we change the allocated parameters ${{a}_{ij}}$ or ${{b}_{j}}$ where $1 < i< n,1 < j < m$ and the output changes to large degree then we conclude that model is very sensitive and not efficient. If the change in output is small then the model is less sensitive and applicable.
We observe that in LPP the output is the optimized cost function or optimal solution and the input are parameters in problem units. So sensitivity analysis determines how an optimal solution in response to problem units. So the correct option is D. \[\]

Note:
We check the sensitivity of an LPP with a simplex table and slack variables. The first two options (A) to allocate resources optimally and (B) to minimize cost operations are the goals of LPP. The third option (C) spell out the relation between primal and dual is the goal of the duality problem in LPP.