Consider the following LPP: \(\begin{array}{ll} \text{Min.} Z= &…
2017
Consider the following LPP:
\(\begin{array}{ll} \text{Min.} Z= & x_{1}+x_{2}+x_{3} \\ \text{Subject to } & 3x_{1}+4x_{3}\leq 5 \\ & 5x_{1}+x_{2}+6x_{3}=7 \\ & 8x_{1}+9x_{3}\geq 2, \\ &x_{1},x_{2},x_{3} \geq 0 \end{array}\)
The standard form of this LPP shall be:
- A.
\(\begin{array}{ll} \text{Min.}Z= & x_{1}+x_{2}+x_{3}+0x_{4}+0x_{5}\\ \text{Subject to} & 3x_{1}+4x_{3}+x_{4}=5;\\ & 5x_{1}+x_{2}+6x_{3}=7;\\ & 8x_{1}+9x_{3}-x_{5} = 2; \\ & x_{1},x_{2},x_{3},x_{4},x_{5}\geq 0 \end{array}\\\) - B.
\(\begin{array}{ll} \text{Min.}Z= & x_{1}+x_{2}+x_{3}+0x_{4}+0x_{5}-1(x_{6})-1(x_{7})\\ \text{Subject to} & 3x_{1}+4x_{3}+x_{4}= 5;\\ & 5x_{1}+x_{2}+6x_{3}+x_{6}=7;\\ & 8x_{1}+9x_{3}-x_{5}+x_{7}= 2;\\ & x_{1}\ \text{to} \ x_{7}\geq 0 \end{array}\) - C.
\(\begin{array}{ll} \text{Min.}Z= & x_{1}+x_{2}+x_{3}+0x_{4}+0x_{5}+0x_{6} \\ \text{Subject to} & 3x_{1}+4x_{3}+x_{4}=5;\\ & 5x_{1}+x_{2}+6x_{3}=7;\\ &8x_{1}+9x_{3}-x_{5}+x_{6} =2;\\ &x_{1}\ \text{to}\ x_{6}\geq 0 \end {array}\\\) - D.
\(\begin{array}{ll} \text{Min.}Z= & x_{1}+x_{2}+x_{3}+ 0x_{4}+ 0x_{5}+ 0x_{6}+ 0x_{7}\\ \text{Subject to} & 3x_{1}+4x_{3}+x_{4}=5;\\ & 5x_{1}+x_{2}+6x_{3}+x_{6}=7\\ & 8x_{1}+9x_{3}-x_{5}+x_{7}= 2; \\ & x_{1}\ \text{to} \ x_{7} \geq 0 \end{array}\)
Show answer & explanation
Correct answer: A
Goal: convert all constraints to equalities and ensure all variables are nonnegative.
Step 1 — Handle each constraint:
For 3x1 + 4x3 ≤ 5: add a slack variable x4 ≥ 0 to get 3x1 + 4x3 + x4 = 5.
For 5x1 + x2 + 6x3 = 7: it is already an equality, so leave it as 5x1 + x2 + 6x3 = 7.
For 8x1 + 9x3 ≥ 2: subtract a surplus variable x5 ≥ 0 to get 8x1 + 9x3 − x5 = 2.
Step 2 — Extend the objective to include the new variables with coefficient 0 (they do not change the original objective function):
Final standard form:
Minimize Z = x1 + x2 + x3 + 0x4 + 0x5
Subject to:
3x1 + 4x3 + x4 = 5
5x1 + x2 + 6x3 = 7
8x1 + 9x3 − x5 = 2
x1, x2, x3, x4, x5 ≥ 0
Notes:
Artificial variables are not part of the standard mathematical form itself; they are introduced later only if needed to find an initial basic feasible solution for the simplex method (using Big M or two-phase).
Do not change the objective coefficients of slack/surplus variables from 0 when writing the standard form; placing arbitrary nonzero penalties there (e.g., −1) is incorrect unless you explicitly apply a method (Big M) and explain the penalty.