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In preparation for the winter season, a clothing company is manufacturing parka and goose overcoats, insulated pants, and gloves. All products are manufactured in four different departments: cutting, insulating, sewing, and packing. The company has received firm orders for its products. The contract stipulates a penalty for undelivered items. The following table provides the pertinent data of the situation.
Department Parka Goose Pants Gloves Capacity
Cutting 0.30 0.30 0.25 0.15 1000
Insulating 0.25 0.35 0.30 0.10 1000
Sewing 0.45 0.50 0.45 0.22 1000
Packaging 0.15 0.15 0.10 0.05 1000
Demand 800 750 600 500
Unit profit 30 40 20 10
Unit Penalty 15 20 10 8
Formulate the Problem as a LP Problem.

Respuesta :

Tundexi

Answer:

Let A represent no of Parka produced

Let B be number of Goose Produced

Let C be number of Number of Pants

Let D be number of Gloves

A', B', C' and D' are shortages in production if any. The range of these variables will be from 0 to Demand.

Out objective is to plan our production to maximize net profit (Profit-Penalty).

Maximize

P = 30A + 40B + 20C + 10D - 15A' - 20B'  - 10C' - 8D'

s.t

0.3A + 0.3B + 0.25C + 0.15D ≤ 1000

0.25A + 0.35B + 0.30C + 0.10D ≤ 1000

0.45A + 0.50B + 0.40C + 0.22D ≤ 1000

0.15A + 0.15B + 0.10C + 0.05D ≤ 1000

0 ≤ A'  ≤ 800

0 ≤ B'  ≤ 750

0 ≤ C'  ≤ 600

0 ≤ D'  ≤ 500

MrRoyal

The LP problem is:

  • Maximize [tex]z= 30x_1 + 40x_2 + 20x_3 +10x_4 - 15s_1 -20s_2 - 10s_3 - 8s_4[/tex]

Subject to

  • [tex]0.3x_1 + 0.3x_2 + 0.25x_3+0.15x_4 \le 1000[/tex]
  • [tex]0.25x_1 + 0.35x_2 + 0.3x_3+0.1x_4 \le 1000[/tex]
  • [tex]0.45x_1 + 0.5x_2 + 0.45x_3+0.22x_4 \le 1000[/tex]
  • [tex]0.15x_1 + 0.15x_2 + 0.1x_3+0.05x_4 \le 1000[/tex]
  • [tex]x_1 + s_1 = 800[/tex], [tex]x_2 + s_2 = 750[/tex], [tex]x_3 + s_3 = 600[/tex], [tex]x_4 + s_4 = 500[/tex]
  • Where: [tex]x_1,x_2,x_3,x_4 \ge 0[/tex]

Represent the products with x1, x2, x3 and x4, and the slack variables with s

From the table entries, we have the unit profit and the unit penalty.

So, the maximized function would be:

Total profit - Total penalty.

This gives

[tex]z= 30x_1 + 40x_2 + 20x_3 +10x_4 - 15s_1 -20s_2 - 10s_3 - 8s_4[/tex]

The constraints for cutting, insulating, sewing and packaging would be:

[tex]0.3x_1 + 0.3x_2 + 0.25x_3+0.15x_4 \le 1000[/tex]

[tex]0.25x_1 + 0.35x_2 + 0.3x_3+0.1x_4 \le 1000[/tex]

[tex]0.45x_1 + 0.5x_2 + 0.45x_3+0.22x_4 \le 1000[/tex]

[tex]0.15x_1 + 0.15x_2 + 0.1x_3+0.05x_4 \le 1000[/tex]

Lastly, the demand entries would be the sum of the product variables, and the slack variables.

So, we have:

[tex]x_1 + s_1 = 800[/tex]

[tex]x_2 + s_2 = 750[/tex]

[tex]x_3 + s_3 = 600[/tex]

[tex]x_4 + s_4 = 500[/tex]

Hence, the LP problem is:

Maximize [tex]z= 30x_1 + 40x_2 + 20x_3 +10x_4 - 15s_1 -20s_2 - 10s_3 - 8s_4[/tex]

Subject to

[tex]0.3x_1 + 0.3x_2 + 0.25x_3+0.15x_4 \le 1000[/tex]

[tex]0.25x_1 + 0.35x_2 + 0.3x_3+0.1x_4 \le 1000[/tex]

[tex]0.45x_1 + 0.5x_2 + 0.45x_3+0.22x_4 \le 1000[/tex]

[tex]0.15x_1 + 0.15x_2 + 0.1x_3+0.05x_4 \le 1000[/tex]

[tex]x_1 + s_1 = 800[/tex], [tex]x_2 + s_2 = 750[/tex], [tex]x_3 + s_3 = 600[/tex], [tex]x_4 + s_4 = 500[/tex]

Where:

[tex]x_1,x_2,x_3,x_4 \ge 0[/tex]

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