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. table 14 computer labor chips equipment costs ($) selling price ($) pea 1 hour 2 5,000 400 apricot 2 hours 5 7,000 900 table 15 machine fixed cost ($) variable cost per unit ($) capacity 1 1,000 20 900 2 920 24 1,000 3 800 16 1,200 4 700 28 1,000

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. table 14 computer labor chips equipment costs ($) selling price ($) pea 1 hour 2 5,000 400 apricot 2 hours 5 7,000 900 table 15 machine fixed cost ($) variable cost per unit ($) capacity 1 1,000 20 900 2 920 24 1,000 3 800 16 1,200 4 700 28 1,000

Answer

Explanation:

Step1: Define decision variables

Let $x_i$ be the number of units produced on machine $i$, where $i = 1,2,3,4$. 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 $x_i$ must not exceed its capacity $C_i$. So, $x_i\leq C_iy_i$ for $i = 1,2,3,4$, where $C_1 = 900,C_2 = 1000,C_3 = 1200,C_4 = 1000$.

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:

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

$x_1\leq900y_1$

$x_2\leq1000y_2$

$x_3\leq1200y_3$

$x_4\leq1000y_4$

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

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

Answer:

The integer programming model is: Minimize $Z = 1000y_1+920y_2 + 800y_3+700y_4+20x_1+24x_2+16x_3+28x_4$ Subject to: $\sum_{i = 1}^{4}x_i=2000$ $x_1\leq900y_1$ $x_2\leq1000y_2$ $x_3\leq1200y_3$ $x_4\leq1000y_4$ $x_i\geq0$ and integer, $i = 1,2,3,4$ $y_i\in{0,1}$, $i = 1,2,3,4$