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

17 a product can be produced on four different machines. each machine has a fixed setup cost, variable production costs per - unit - processed, and a production capacity given in table 15. a total of 2,000 units of the product must be produced. formulate an ip whose solution will tell us how to minimize total costs.\n\ntable 14\n\n| computer | labor | chips | equipment costs ($) | selling price ($) |\n| ---- | ---- | ---- | ---- | ---- |\n| pear | 1 hour | 2 | 5,000 | 400 |\n| apricot | 2 hours | 5 | 7,000 | 900 |\n\ntable 15\n\n| machine | fixed cost ($) | variable cost per unit ($) | capacity |\n| ---- | ---- | ---- | ---- |\n| 1 | 1,000 | 20 | 900 |\n| 2 | 920 | 24 | 1,000 |\n| 3 | 800 | 16 | 1,200 |\n| 4 | 700 | 28 | 1,600 |

17 a product can be produced on four different machines. each machine has a fixed setup cost, variable production costs per - unit - processed, and a production capacity given in table 15. a total of 2,000 units of the product must be produced. formulate an ip whose solution will tell us how to minimize total costs.\n\ntable 14\n\n| computer | labor | chips | equipment costs ($) | selling price ($) |\n| ---- | ---- | ---- | ---- | ---- |\n| pear | 1 hour | 2 | 5,000 | 400 |\n| apricot | 2 hours | 5 | 7,000 | 900 |\n\ntable 15\n\n| machine | fixed cost ($) | variable cost per unit ($) | capacity |\n| ---- | ---- | ---- | ---- |\n| 1 | 1,000 | 20 | 900 |\n| 2 | 920 | 24 | 1,000 |\n| 3 | 800 | 16 | 1,200 |\n| 4 | 700 | 28 | 1,600 |

Answer

Explanation:

Step1: Define decision variables

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

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

Production quantity constraint

The total number of units produced must be 2000. So, $\sum_{i = 1}^{4}x_i=2000$.

Capacity constraints

For each machine $i$, the number of units produced on it cannot exceed its capacity $C_i$. So, $x_i\leq C_iy_i$, where $C_1 = 900,C_2=1000,C_3 = 1200,C_4=1600$. Also, $x_i\geq0$ and $x_i$ is an integer, and $y_i\in{0,1}$ for $i = 1,2,3,4$.

The integer - programming (IP) formulation is:

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

Subject to:

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

$x_1\leq900y_1$

$x_2\leq1000y_2$

$x_3\leq1200y_3$

$x_4\leq1600y_4$

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

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

Answer:

The IP formulation is: Minimize $Z = 1000y_1+920y_2 + 800y_3+700y_4+20x_1+24x_2+16x_3+28x_4$ subject to $\sum_{i = 1}^{4}x_i=2000$, $x_1\leq900y_1$, $x_2\leq1000y_2$, $x_3\leq1200y_3$, $x_4\leq1600y_4$, $x_i\geq0$, $x_i$ is an integer, $y_i\in{0,1}$ for $i = 1,2,3,4$.