Skip to content

LP problems with two variables – Type A

Catalogue of LP problems · Problem catalogue (PDF)

LP Problem \(P0\)

\[ \begin{array}{rrrrrrrrrrrrr} {\max} & 2 \, {x_1} & {+} & 4 \, {x_2} \\[2ex] \text{subject to} & 2 \, {x_1} & {+} & 3 \, {x_2} & {\le} & 4 \\[2ex] & {x_1,} & & {x_2} & { \ge} & { 0 } \end{array} \]

Figure 1

LP Problem \(P1\)

\[ \begin{array}{rrrrrrrrrrrrr} {\max} & {x_1} & {+} & 8 \, {x_2} \\[2ex] \text{subject to} & {3} \, {x_1} & {+} & {6} \, {x_2} & {\le} & {16} \\[2ex] & {x_1,} & & {x_2} & { \ge} & {0 } \end{array} \]

Figure 2

LP Problem \(P2\)

\[ \begin{array}{rrrrrrrrrrrrr} {\max} & {x_1} & {+} & 3 \, {x_2} \\[2ex] \text{subject to} & {-}4 \, {x_1} & {+} & 10 \, {x_2} & {\le} & 35 \\[2ex] & 3 \, {x_1} & {+} & \, {x_2} & {\le} & 12 \\[2ex] & {x_1,} & & {x_2} & { \ge} & { 0 } \end{array} \]

Figure 3

LP Problem \(P3\)

\[ \begin{array}{rrrrrrrrrrrrr} {\max} & {2} \, {x_1} & {+} & {4} \, {x_2} \\[2ex] \text{subject to} & {-} \, {x_1} & {+} & \, {x_2} & {\le} & {2} \\[2ex] & {8} \, {x_1} & {+} & {2} \, {x_2} & {\le} & {19} \\[2ex] & {x_1,} & & {x_2} & { \ge} & { 0 } \end{array} \]

Figure 4

LP Problem \(P4\)

\[ \begin{array}{rrrrrrrrrrrrr} {\max} & 5 {x_1} & {+} & 3 \, {x_2} \\[2ex] \text{subject to} & 2 \, {x_1} & {+} & \, {x_2} & {\le} & 14 \\[2ex] & \, {x_1} & {+} & 8 \, {x_2} & {\le} & 32 \\[2ex] & {x_1,} & & {x_2} & { \ge} & { 0 } \end{array} \]

Figure 5

LP Problem \(P5\)

\[ \begin{array}{rrrrrrrrrrrrr} {\max} & 2 {x_1} & {+} & 3 \, {x_2} \\[2ex] \text{subject to} & 6 \, {x_1} & {+} & 21 \, {x_2} & {\le} & 77 \\[2ex] & 8 \, {x_1} & {+} & 3 \, {x_2} & {\le} & 36 \\[2ex] & {x_1,} & & {x_2} & { \ge} & { 0 } \end{array} \]

Figure 6

ILP problem \(P6\)

\[ \begin{array}{rrrrrrrrrrrrr} {\max} & {x_1} & {+} & \, {x_2} \\[2ex] \text{subject to} & {4} \, {x_1} & {+} & {15} \, {x_2} & {\le} & {70} \\[2ex] & {12} \, {x_1} & {+} & {3} \, {x_2} & {\le} & {52} \\[2ex] & {x_1,} & & {x_2} & {\in} & {\mathbb{Z}_{\ge 0}} \end{array} \]

Figure 7

LP Problem \(P7\)

\[ \begin{array}{rrrrrrrrrrrrr} {\max} & 3 \, {x_1} & {+} & {x_2} \\[2ex] \text{subject to} & {x_1} & {+} & {4} \, {x_2} & {\le} & {12} \\[2ex] & 2 \, {x_1} & {+} & \, {x_2} & {\le} & 5 \\[2ex] & {x_1,} & & {x_2} & {\ge} & {0} \end{array} \]

Figure 8

LP Problem \(P8\)

\[ \begin{array}{rrrrrrrrrrrrr} {\max} & \, {x_1} & {+} & 2 \, {x_2} \\[2ex] \text{subject to} & {-} {x_1} & {+} & {2} \, {x_2} & {\le} & {2} \\[2ex] & \, {x_1} & {+} & \, {x_2} & {\le} & 3 \\[2ex] & \, {x_1,} & & {x_2} & { \ge } & { 0 } \end{array} \]

Figure 9

LP Problem \(P9\)

\[ \begin{array}{rrrrrrrrrrrrr} {\max} & 2 \, {x_1} & {+} & {x_2} \\[2ex] \text{subject to} & {-} \, {x_1} & {+} & \, {x_2} & {\le} & 1 \\[2ex] & {6} \, {x_1} & {+} & {4} \, {x_2} & {\le} & {39} \\[2ex] & {x_1,} & & {x_2} & { \ge} & {0} \end{array} \]

Figure 10

LP Problem \(P10\)

\[ \begin{array}{rrrrrrrrrrrrr} {\max} & 5 \, {x_1} & {+} & 2 \, {x_2} \\[2ex] \text{subject to} & {-} \, {x_1} & {+} & 3 \, {x_2} & {\le} & 12 \\[2ex] & {x_1} & {+} & 2 \, {x_2} & {\le} & 12 \\[2ex] & {3} \, {x_1} & {-} & 2 \, {x_2} & {\le} & {8} \\[2ex] & {x_1,} & & {x_2} & { \ge} & {0} \end{array} \]

Figure 11