What Is the Smallest Value of the Objective Function Coefficient

question 67

Essay

What is the smallest value of the objective function coefficient X₁ can assume without changing the optimal solution?
$$\begin{array} { l l } \operatorname { MAX } : & 7 \mathbf { X } _ { 1 } + 4 \mathbf { X } _ { \mathbf { 2 } } \\\text { Subject ta: } & 2 \mathbf { X } _ { 1 } + \mathbf { X } _ { \mathbf { 2 } } \leq 16 \\& \mathbf { X } _ { 1 } + \mathbf { X } _ { \mathbf { 2 } } \leq 10 \\& 2 \mathbf { X } _ { 1 } + 5 \mathbf { X } _ { \mathbf { 2 } } \leq 40 \\& \mathbf { X } _ { 1 } \mathbf { X } _ { \mathbf { 2 } } \geq \mathbf {0 }\end{array}$$ $\text {Changing Calls}$
$\begin{array}{llrrrrr}&&\text { Final} &\text {Reduced }&\text {Objective }&\text {Allowable}&\text { Allowable }\\\text { Cell } & \text { Name } & \text { Value } & \text { Cost } & \text { Coefficient } & \text { Increase } & \text { Decrease } \\\hline \$ \mathrm{~B} \$ 4 & \text { Number to make: } \mathrm{X} 1 &6& 0 & 7& 1& 3\\\$ \mathrm{C} \$ 4 & \text { Number to make: } \mathrm{X} 2 & 4 & 0 &4 &3 & 0.5\\\end{array}$

$\text {Constraints}$
$\begin{array}{llrrrrr}&&\text { Final } &\text {Shadow} &\text { Constraint } &\text {Allowable} &\text { Allowable }\\\text { Cell } & \text { Name } & \text { Value } & \text { Price } & \text { R.H. Side } & \text { Increase } & \text { Decrease } \\\hline \$ \mathrm{D} \$ 8 & \text { Used } & 16&3 & 16&4&2.67 \\\$ \mathrm{D} \$ 9 & \text { Used } & 10 & 1&10 & 1 & 2 \\\$ \mathrm{D} \$ 10 & \text { Used } & 32& 0 & 40 & 1 E+30 & 8\end{array}$

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Definitions:

Causality

The relationship between cause and effect, where one event leads to the outcome of another.

Vulnerability

The quality or state of being exposed to the possibility of being attacked or harmed, either physically or emotionally.

Suicide

Voluntarily causing one's own demise.

Loss of a Loved One

The experience of grief and mourning following the death of someone significant in one's life.