An objective function $Z$ of primal variable $\left(x_1,x_2\right)$ is described. $K_1,K_2,K_3,K_4$ are the dual variables.

\text{Min }Z=0.07x_1+0.05x_2

Subject to:

$0.1x_1\ge 0.4$

$0.1x_2\ge 0.6$

$0.1x_1+0.2x_2\ge 2.0$

$0.2x_1+0.1x_2\ge 1.8$

$x_1,x_2\ge 0$

$W$ is the objective function of dual of $Z$. Which option is correct?

The correct option is:
$$\text{Max W =}0.4K_1+0.6K_2+2.0K_3+1.8K_4$$
Subject to:
$$0.1K_1+0.1K_3+0.2K_4\le 0.07$$
$$0.1K_2+0.2K_3+0.1K_4\le 0.05$$
where:
$$K_1,K_2,K_3,K_4\ge 0$$

Step-by-Step Explanation:

1. Understanding the Primal Linear Programming Problem (LPP)
We are given a primal minimization problem with two primal decision variables, $x_1$ and $x_2$:
$$\text{Minimize}Z=0.07x_1+0.05x_2$$
Subject to the constraints:
Constraint 1: $0.1x_1+0x_2\ge 0.4$ (associated with dual variable $K_1$)
Constraint 2: $0x_1+0.1x_2\ge 0.6$ (associated with dual variable $K_2$)
Constraint 3: $0.1x_1+0.2x_2\ge 2.0$ (associated with dual variable $K_3$)
Constraint 4: $0.2x_1+0.1x_2\ge 1.8$ (associated with dual variable $K_4$)
And the non-negativity restrictions: $x_1,x_2\ge 0$

2. Constructing the Dual Objective Function ($W$)
For a primal minimization LPP, the corresponding dual LPP is a maximization problem.
The coefficients of the dual objective function $W$ are the right-hand side constants of the primal constraints:
The right-hand sides are: $0.4$, $0.6$, $2.0$, and $1.8$.
Therefore, the dual objective function is:
$$\text{Maximize}W=0.4K_1+0.6K_2+2.0K_3+1.8K_4$$

3. Determining the Dual Constraints
Since the primal has two variables, $x_1$ and $x_2$, the dual will have two constraints.
For each primal variable $x_j$, we look at its coefficients across all constraints to form the $j$-th dual constraint:

• First dual constraint (corresponding to primal variable $x_1$):
  The coefficients of $x_1$ in the four primal constraints are: $0.1$ (in constraint 1), $0$ (in constraint 2), $0.1$ (in constraint 3), and $0.2$ (in constraint 4).
  Since the primal is a minimization problem with "$\ge$" constraints, the dual constraints will be "$\le$" constraints, bounded by the coefficient of $x_1$ in the primal objective function, which is $0.07$.
  Thus:
  $$0.1K_1+0K_2+0.1K_3+0.2K_4\le 0.07$$
  Which simplifies to:
  $$0.1K_1+0.1K_3+0.2K_4\le 0.07$$
• Second dual constraint (corresponding to primal variable $x_2$):
  The coefficients of $x_2$ in the four primal constraints are: $0$ (in constraint 1), $0.1$ (in constraint 2), $0.2$ (in constraint 3), and $0.1$ (in constraint 4).
  This constraint is bounded by the coefficient of $x_2$ in the primal objective function, which is $0.05$.
  Thus:
  $$0K_1+0.1K_2+0.2K_3+0.1K_4\le 0.05$$
  Which simplifies to:
  $$0.1K_2+0.2K_3+0.1K_4\le 0.05$$

4. Non-negativity Conditions
Since the primal constraints are all inequalities ($\ge$), the dual variables $K_1,K_2,K_3,K_4$ must be non-negative:
$$K_1,K_2,K_3,K_4\ge 0$$