17 a product can be produced on four different machines. each machine has a fixed setup cost, variable…

Explanation:

Step1: Define decision variables

Let $x_i$ be the number of units produced on machine $i$ ($i = 1,2,3,4$), and $y_i$ be a binary variable such that $y_i=1$ if machine $i$ is used and $y_i = 0$ if it is not used.

Step2: Define the objective function

The total - cost function $Z$ is the sum of the fixed - setup costs and the variable production costs. The fixed - setup cost for machine $i$ is $F_i$ and the variable cost per unit is $v_i$. So, $Z=\sum_{i = 1}^{4}F_iy_i+\sum_{i = 1}^{4}v_ix_i$, where $F_1 = 1000,F_2=920,F_3 = 800,F_4=700$ and $v_1 = 20,v_2=24,v_3 = 16,v_4=28$.

Step3: Define the constraints

1. Production - quantity constraint: The total number of units produced must be 2000. So, $\sum_{i = 1}^{4}x_i=2000$.
2. Capacity constraints: For each machine $i$, $x_i\leq C_iy_i$, where $C_1 = 900,C_2=1000,C_3 = 1200,C_4=1600$.
3. Non - negativity and integrality constraints: $x_i\geq0$ and integer for $i = 1,2,3,4$, and $y_i\in{0,1}$ for $i = 1,2,3,4$.

The integer programming (IP) model is:

Minimize $Z = 1000y_1+920y_2 + 800y_3+700y_4+20x_1+24x_2+16x_3+28x_4$

Subject to:

$\sum_{i = 1}^{4}x_i=2000$

$x_1\leq900y_1$

$x_2\leq1000y_2$

$x_3\leq1200y_3$

$x_4\leq1600y_4$

$x_i\geq0$ and integer, $i = 1,2,3,4$

$y_i\in{0,1}$, $i = 1,2,3,4$

Answer:

The formulated integer programming model is:

Minimize $Z = 1000y_1+920y_2 + 800y_3+700y_4+20x_1+24x_2+16x_3+28x_4$

Subject to:

$\sum_{i = 1}^{4}x_i=2000$

$x_1\leq900y_1$

$x_2\leq1000y_2$

$x_3\leq1200y_3$

$x_4\leq1600y_4$

$x_i\geq0$ and integer, $i = 1,2,3,4$

$y_i\in{0,1}$, $i = 1,2,3,4$