a maximisation LP, solved (simplex method). Operations Research (OR)

1. Maximisation Problem
LPP

2. Product mix problem. A company manufactures two different types of
products, P1 and P2. Each product requires processing on milling
machine and drilling machine. But each type of machine has limited
hours available per week. The net profit per unit of the products,
resource requirements of the products and availability of resources are
summarized below. Develop a linear programming model and solve it to
determine the optimal production volume of each of the products such
that the profit is maximized subject to the availability of machine hours.
Machine type Processing time (hours) Machine hours
available per weekProduct P1 Product P2
Milling machine 2 5 200
Drilling machine 4 2 240
Profit/unit (Rs.) 250 400

3. 𝑴𝒂𝒙𝒊𝒎𝒊𝒔𝒆 𝒛 = 𝟐𝟓𝟎𝒙 𝟏 + 𝟒𝟎𝟎𝒙 𝟐
𝒔𝒖𝒃𝒋𝒆𝒄𝒕 𝒕𝒐 𝒕𝒉𝒆 𝒄𝒐𝒏𝒔𝒕𝒓𝒂𝒊𝒏𝒕𝒔:
𝟐𝒙 𝟏 + 𝟓𝒙 𝟐 ≤ 𝟐𝟎𝟎
𝟒𝒙 𝟏 + 𝟐𝒙 𝟐 ≤ 𝟐𝟒𝟎
𝒙 𝟏 ≥ 𝟎
𝒙 𝟐 ≥ 𝟎

4. 𝑰𝒏𝒕𝒓𝒐𝒅𝒖𝒄𝒊𝒏𝒈 𝑺𝒍𝒂𝒄𝒌 𝒗𝒂𝒓𝒊𝒂𝒃𝒍𝒆𝒔
𝑴𝒂𝒙𝒊𝒎𝒊𝒔𝒆 𝒛 = 𝟐𝟓𝟎𝒙 𝟏 + 𝟒𝟎𝟎𝒙 𝟐 + 𝟎𝑺 𝟏 + 𝟎𝑺 𝟐
𝒔𝒖𝒃𝒋𝒆𝒄𝒕 𝒕𝒐 𝒕𝒉𝒆 𝒄𝒐𝒏𝒔𝒕𝒓𝒂𝒊𝒏𝒕𝒔:
𝟐𝒙 𝟏 + 𝟓𝒙 𝟐 + 𝑺 𝟏 + 𝟎𝑺 𝟐 ≤ 𝟐𝟎𝟎
𝟒𝒙 𝟏 + 𝟐𝒙 𝟐 + 𝟎𝑺 𝟏 + 𝑺 𝟐 ≤ 𝟐𝟒𝟎
𝒙 𝟏 ≥ 𝟎
𝒙 𝟐 ≥ 𝟎
𝑺 𝟏 ≥ 𝟎
𝑺 𝟐 ≥ 𝟎

5. Initial Simplex Table

6. 𝑪𝒋 𝟐𝟓𝟎 𝟒𝟎𝟎 𝟎 𝟎
𝑪 𝒃 𝑩𝑽 𝑺𝑽 𝒙 𝟏 𝒙 𝟐 𝑺 𝟏 𝑺 𝟐
𝟎 𝑺 𝟏 𝟐𝟎𝟎 𝟐 𝟓 𝟏 𝟎
𝟐𝟎𝟎
𝟓
= 𝟒𝟎 ←
𝟎 𝑺 𝟐 𝟐𝟒𝟎 𝟒 𝟐 𝟎 𝟏
𝟐𝟒𝟎
𝟐
= 𝟏𝟐𝟎
𝒁𝒋 𝟎 𝟎 𝟎 𝟎 𝟎
𝑪𝒋 − 𝒁𝒋 𝟐𝟓𝟎 𝟒𝟎𝟎 ↑ 𝟎 𝟎

7. Second Simplex Table

8. 𝑪𝒋 𝟐𝟓𝟎 𝟒𝟎𝟎 𝟎 𝟎
𝑪 𝒃 𝑩𝑽 𝑺𝑽 𝒙 𝟏 𝒙 𝟐 𝑺 𝟏 𝑺 𝟐
𝟒𝟎𝟎 𝒙 𝟐 𝟒𝟎
𝟐
𝟓
𝟏
𝟏
𝟓
𝟎 𝑹 𝟏 =
𝑹 𝟏
𝟓
𝟒𝟎
𝟐
𝟓
= 𝟏𝟎𝟎
𝟎 𝑺 𝟐 𝟏𝟔𝟎
𝟏𝟔
𝟓
𝟎 −
𝟐
𝟓
𝟏 𝑹 𝟐 = 𝑹 𝟐 − 𝟐𝑹 𝟏
𝟏𝟔𝟎
𝟏𝟔
𝟓
= 𝟓𝟎 ←
𝒁𝒋 𝟏𝟔𝟎𝟎𝟎 𝟏𝟔𝟎 𝟒𝟎𝟎 𝟖𝟎 𝟎
𝑪𝒋 − 𝒁𝒋 𝟗𝟎 ↑ 𝟎 −𝟖𝟎 𝟎

9. Final Simplex Table

10. 𝑪𝒋 𝟐𝟓𝟎 𝟒𝟎𝟎 𝟎 𝟎
𝑪 𝒃 𝑩𝑽 𝑺𝑽 𝒙 𝟏 𝒙 𝟐 𝑺 𝟏 𝑺 𝟐
𝟒𝟎𝟎 𝒙 𝟐 𝟐𝟎 𝟎 𝟏
𝟏
𝟒
−
𝟏
𝟖
𝑹 𝟏 = 𝑹 𝟏 −
𝟐
𝟓
𝑹 𝟐
𝟐𝟓𝟎 𝒙 𝟏 𝟓𝟎 𝟏 𝟎 −
𝟏
𝟖
𝟓
𝟏𝟔
𝑹 𝟐 =
𝑹 𝟐
𝟏𝟔
𝟓
𝒁𝒋 𝟐𝟎𝟓𝟎𝟎 𝟐𝟓𝟎 𝟒𝟎𝟎
𝟐𝟕𝟓
𝟒
𝟐𝟐𝟓
𝟖
𝑪𝒋 − 𝒁𝒋 𝟎 𝟎 −
𝟐𝟕𝟓
𝟒
−
𝟐𝟐𝟓
𝟖

11. 𝑨𝒏𝒔: 𝒙 𝟏 = 𝟓𝟎, 𝒙 𝟐 = 𝟐𝟎, 𝒛 = 𝟐𝟎𝟓𝟎𝟎