LPP formulation. operations research

1. LP Formulation

2. Two products, namely P1 and P2 are being manufactured.
Each product has to be processed through two machines
M1 and M2. One unit of product P1 consumes 4 hours of
time on M1 and 2 hours of time on M2. Similarly, one unit
of P2 consumes 2 hours of time on M1 and 4 hours of
time on M2. 60 hours of time is available on M1 and 48
hours on M2. The per unit contribution margin of P1 is 8
and of P2 is 6. Determine the number of units of P1 and
P2 to be manufactured so as to maximise total
contribution.

3. Products M1 M2 Contribution
Margin
P1 4hrs 2hrs Rs.8
P2 2hrs 4hrs Rs.6
Total Machine
hours
60hrs 48hrs

4. 𝑶𝒃𝒋𝒆𝒄𝒕𝒊𝒗𝒆: 𝑻𝒐 𝒎𝒂𝒙𝒊𝒎𝒊𝒔𝒆 𝒕𝒉𝒆 𝒕𝒐𝒕𝒂𝒍 𝒄𝒐𝒏𝒕𝒓𝒊𝒃𝒖𝒕𝒊𝒐𝒏 𝒎𝒂𝒓𝒈𝒊𝒏
𝑪𝒐𝒏𝒔𝒕𝒓𝒂𝒊𝒏𝒕𝒔: 𝑨𝒗𝒂𝒊𝒍𝒂𝒃𝒊𝒍𝒊𝒕𝒚 𝒐𝒇 𝒎𝒂𝒄𝒉𝒊𝒏𝒆 𝒉𝒐𝒖𝒓𝒔
𝑳𝒆𝒕 𝒖𝒔 𝒂𝒔𝒔𝒖𝒎𝒆 𝒕𝒉𝒂𝒕 𝒙 𝟏 𝒖𝒏𝒊𝒕𝒔 𝒐𝒇 𝑷𝟏 𝒂𝒏𝒅 𝒙 𝟐 𝒖𝒏𝒊𝒕𝒔 𝒐𝒇 𝑷𝟐 𝒂𝒓𝒆
𝒑𝒓𝒐𝒅𝒖𝒄𝒆𝒅
𝑶𝒃𝒆𝒋𝒄𝒕𝒊𝒗𝒆 𝒇𝒖𝒏𝒄𝒕𝒊𝒐𝒏:
𝑴𝒂𝒙𝒊𝒎𝒊𝒔𝒆 𝒛 = 𝟖𝒙 𝟏 + 𝟔𝒙 𝟐
𝑴𝒂𝒄𝒉𝒊𝒏𝒆 𝑴𝟏 𝒄𝒐𝒏𝒔𝒕𝒓𝒂𝒊𝒏𝒕:
𝟒𝒙 𝟏 + 𝟐𝒙 𝟐 ≤ 𝟔𝟎
𝑴𝒂𝒄𝒉𝒊𝒏𝒆 𝑴𝟐 𝒄𝒐𝒏𝒔𝒕𝒓𝒂𝒊𝒏𝒕:
𝟐𝒙 𝟏 + 𝟒𝒙 𝟐 ≤ 𝟒𝟖
𝑵𝒐𝒏 − 𝒏𝒆𝒈𝒂𝒕𝒊𝒗𝒊𝒕𝒚 𝒄𝒐𝒏𝒔𝒕𝒓𝒂𝒊𝒏𝒕𝒔:
𝒙 𝟏 ≥ 𝟎
𝒙 𝟐 ≥ 𝟎

5. ∴ 𝑻𝒉𝒆 𝒇𝒐𝒓𝒎𝒖𝒍𝒂𝒕𝒆𝒅 𝒑𝒓𝒐𝒃𝒍𝒆𝒎 𝒄𝒂𝒏 𝒃𝒆 𝒔𝒕𝒂𝒕𝒆𝒅 𝒂𝒔:
𝑴𝒂𝒙𝒊𝒎𝒊𝒔𝒆 𝒛 = 𝟖𝒙 𝟏 + 𝟔𝒙 𝟐
𝒔𝒖𝒃𝒋𝒆𝒄𝒕 𝒕𝒐 𝒕𝒉𝒆 𝒄𝒐𝒏𝒔𝒕𝒓𝒂𝒊𝒏𝒕𝒔:
𝟒𝒙 𝟏 + 𝟐𝒙 𝟐 ≤ 𝟔𝟎
𝟐𝒙 𝟏 + 𝟒𝒙 𝟐 ≤ 𝟒𝟖
𝒙 𝟏 ≥ 𝟎
𝒙 𝟐 ≥ 𝟎