Interactive Linear Programming: Feasible Region and Optimum Point

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2020-09-09 05:24

This GeoGebra applet interactively demonstrates how a linear programming problem builds a feasible region from linear inequalities and how the objective function reaches its optimum (maximum/minimum) at a particular point of that region. It centers on key ideas of linear programming and linear optimization: constraint-induced half-planes, the polygonal feasible set, and geometric reasoning for finding the optimal value and location. The construction uses GeoGebra commands and elements reflected in the provided construct summary: Vertex to define key vertices of the feasible polygon, OrthogonalVector to create direction/normal relationships used for constraint boundaries, If to handle conditional logic during interaction, RemoveUndefined to clean up objects that may become invalid, Textfield to let users enter or vary parameters, and Max to compute the optimal objective value. The listed objects (e.g., points and line/constraint expressions) together support identifying the optimum among candidate corner points. Suitable for advanced high-school learners, contest preparation, and introductory college courses on optimization and applied mathematics where visual geometry helps explain algebraic linear programs.

GeoGebra Commands & Parameters

Vertex(g_7) → N_1
Vertex(g) → A, B, C, D, E, F, G, H, I, J
Vertex(g_1) → K, L, M, N, O, P
Vertex(g_2) → Q, R, S, T, U, V
Vertex(g_4) → E_1, F_1, G_1
Vertex(g_3) → W, Z, A_1, B_1, C_1, D_1
Vertex(g_5) → H_1, I_1, J_1
Vertex(g_6) → K_1, L_1, M_1
If(Ineq1 ≟ true ∧ Ineq2 ≟ true ∧ Ineq3 ≟ true, {If[IsInRegion[A, g], A], If[IsInRegion[B, g] ∧ B ≠ A, B], If[IsInRegion[C, g] ∧ C ≠ A ∧ C ≠ B, C], If[IsInRegion[D, g] ∧ D ≠ A ∧ D ≠ B ∧ D ≠ C, D], If[IsInRegion[E, g] ∧ E ≠ A ∧ E ≠ B ∧ E ≠ C ∧ E ≠ D, E], If[IsInRegion[F, g] ∧ F ≠ A ∧ F ≠ B ∧ F ≠ C ∧ F ≠ D ∧ F ≠ E, F], If[IsInRegion[G, g] ∧ G ≠ A ∧ G ≠ B ∧ G ≠ C ∧ G ≠ D ∧ G ≠ E ∧ G ≠ F, G], If[IsInRegion[H, g] ∧ H ≠ A ∧ H ≠ B ∧ H ≠ C ∧ H ≠ D ∧ H ≠ E ∧ H ≠ (F * H) ≠ G, H], If[IsInRegion[I, g] ∧ I ≠ A ∧ I ≠ B ∧ I ≠ C ∧ I ≠ D ∧ I ≠ E ∧ I ≠ F ∧ I ≠ G ∧ I ≠ H, I], If[IsInRegion[J, g] ∧ J ≠ A ∧ J ≠ B ∧ J ≠ C ∧ J ≠ D ∧ J ≠ E ∧ J ≠ F ∧ J ≠ G ∧ J ≠ H ∧ J ≠ I, J]}, If[Ineq1 ≟ true ∧ Ineq2 ≟ true ∧ Ineq3 ≟ false, {If[IsInRegion[K, g_1], K], If[IsInRegion[L, g_1] ∧ L ≠ K, L], If[IsInRegion[M, g_1] ∧ M ≠ K ∧ M ≠ L, M], If[IsInRegion[N, g_1] ∧ N ≠ K ∧ N ≠ L ∧ N ≠ M, N], If[IsInRegion[O, g_1] ∧ O ≠ K ∧ O ≠ L ∧ O ≠ M ∧ O ≠ N, O], If[IsInRegion[P, g_1] ∧ P ≠ K ∧ P ≠ L ∧ P ≠ M ∧ P ≠ N ∧ P ≠ O, P]}, If[Ineq1 ≟ true ∧ Ineq2 ≟ false ∧ Ineq3 ≟ true, {If[IsInRegion[Q, g_2], Q], If[IsInRegion[R, g_2] ∧ R ≠ Q, R], If[IsInRegion[S, g_2] ∧ S ≠ Q ∧ S ≠ R, S], If[IsInRegion[T, g_2] ∧ T ≠ Q ∧ T ≠ R ∧ T ≠ S, T], If[IsInRegion[U, g_2] ∧ U ≠ Q ∧ U ≠ R ∧ U ≠ S ∧ U ≠ T, U], If[IsInRegion[V, g_2] ∧ V ≠ Q ∧ V ≠ R ∧ V ≠ S ∧ V ≠ T ∧ V ≠ U, V]}, If[Ineq1 ≟ false ∧ Ineq2 ≟ true ∧ Ineq3 ≟ true, {If[IsInRegion[W, g_3], W], If[IsInRegion[Z, g_3] ∧ Z ≠ W, Z], If[IsInRegion[A_1, g_3] ∧ A_1 ≠ W ∧ A_1 ≠ Z, A_1], If[IsInRegion[B_1, g_3] ∧ B_1 ≠ W ∧ B_1 ≠ Z ∧ B_1 ≠ A_1, B_1], If[IsInRegion[C_1, g_3] ∧ C_1 ≠ W ∧ C_1 ≠ Z ∧ C_1 ≠ A_1 ∧ C_1 ≠ B_1, C_1], If[IsInRegion[D_1, g_3] ∧ D_1 ≠ W ∧ D_1 ≠ Z ∧ D_1 ≠ A_1 ∧ D_1 ≠ B_1 ∧ D_1 ≠ C_1, D_1]}, If[Ineq1 ≟ true ∧ Ineq2 ≟ false ∧ Ineq3 ≟ false, {If[IsInRegion[E_1, g_4], E_1], If[IsInRegion[F_1, g_4] ∧ F_1 ≠ E_1, F_1], If[IsInRegion[G_1, g_4] ∧ G_1 ≠ E_1 ∧ G_1 ≠ F_1, G_1]}, If[Ineq1 ≟ false ∧ Ineq2 ≟ true ∧ Ineq3 ≟ false, {If[IsInRegion[H_1, g_5], H_1], If[IsInRegion[I_1, g_5] ∧ I_1 ≠ H_1, I_1], If[IsInRegion[J_1, g_5] ∧ J_1 ≠ H_1 ∧ J_1 ≠ I_1, J_1]}, If[Ineq1 ≟ false ∧ Ineq2 ≟ false ∧ Ineq3 ≟ true, {If[IsInRegion[K_1, g_6], K_1], If[IsInRegion[L_1, g_6] ∧ L_1 ≠ K_1, L_1], If[IsInRegion[M_1, g_6] ∧ M_1 ≠ K_1 ∧ M_1 ≠ L_1, M_1]}, If[Ineq1 ≟ false ∧ Ineq2 ≟ false ∧ Ineq3 ≟ false, {If[IsInRegion[N_1, g_7], N_1]}]]]]]]]) → l1
RemoveUndefined(l1) → l2
Textfield(a) → InputBox5
Textfield(b) → InputBox6
Textfield(c) → InputBox7
Textfield(d) → InputBox10
Textfield(e) → InputBox11
Textfield(f) → InputBox12
OrthogonalVector(l) → u
If(A4(1) > A4(2) ∧ A4(1) > A4(3) ∧ A4(1) > A4(4) ∧ A4(1) > A4(5), A4(1), If[A4(2) > A4(1) ∧ A4(2) > A4(3) ∧ A4(2) > A4(4) ∧ A4(2) > A4(5), A4(2), If[A4(3) > A4(1) ∧ A4(3) > A4(2) ∧ A4(3) > A4(4) ∧ A4(3) > A4(5), A4(3), If[A4(4) > A4(1) ∧ A4(4) > A4(2) ∧ A4(4) > A4(3) ∧ A4(4) > A4(5), A4(4), If[A4(5) > A4(1) ∧ A4(5) > A4(2) ∧ A4(5) > A4(3) ∧ A4(5) > A4(4), A4(5), If[A4(1) > A4(2) ∧ A4(1) > A4(3) ∧ A4(1) > A4(4), A4(1), If[A4(2) > A4(1) ∧ A4(2) > A4(3) ∧ A4(2) > A4(4), A4(2), If[A4(3) > A4(1) ∧ A4(3) > A4(2) ∧ A4(3) > A4(4), A4(3), If[A4(4) > A4(1) ∧ A4(4) > A4(2) ∧ A4(4) > A4(3), A4(4), If[A4(1) > A4(2) ∧ A4(1) > A4(3), A4(1), If[A4(2) > A4(1) ∧ A4(2) > A4(3), A4(2), If[A4(3) > A4(1) ∧ A4(3) > A4(2), A4(3), If[A4(1) > A4(2), A4(1), If[A4(2) > A4(1), A4(2), A4(1)]]]]]]]]]]]]]) → Max
Max(A4) → h
If(A4(1) < A4(2) ∧ A4(1) < A4(3) ∧ A4(1) < A4(4) ∧ A4(1) < A4(5), A4(1), If[A4(2) < A4(1) ∧ A4(2) < A4(3) ∧ A4(2) < A4(4) ∧ A4(2) < A4(5), A4(2), If[A4(3) < A4(1) ∧ A4(3) < A4(2) ∧ A4(3) < A4(4) ∧ A4(3) < A4(5), A4(3), If[A4(4) < A4(1) ∧ A4(4) < A4(2) ∧ A4(4) < A4(3) ∧ A4(4) < A4(5), A4(4), If[A4(5) < A4(1) ∧ A4(5) < A4(2) ∧ A4(5) < A4(3) ∧ A4(5) < A4(4), A4(5), If[A4(1) < A4(2) ∧ A4(1) < A4(3) ∧ A4(1) < A4(4), A4(1), If[A4(2) < A4(1) ∧ A4(2) < A4(3) ∧ A4(2) < A4(4), A4(2), If[A4(3) < A4(1) ∧ A4(3) < A4(2) ∧ A4(3) < A4(4), A4(3), If[A4(4) < A4(1) ∧ A4(4) < A4(2) ∧ A4(4) < A4(3), A4(4), If[A4(1) < A4(2) ∧ A4(1) < A4(3), A4(1), If[A4(2) < A4(1) ∧ A4(2) < A4(3), A4(2), If[A4(3) < A4(1) ∧ A4(3) < A4(2), A4(3), If[A4(1) < A4(2), A4(1), If[A4(2) < A4(1), A4(2), A4(1)]]]]]]]]]]]]]) → Min
If(Opt ≟ 1 ∧ Max ≟ A4(1) ≟ A4(2), "Segment " + ((((A2(1) * " - ") * A2(2)) * ": ")) + Max, If[Opt ≟ 1 ∧ Max ≟ A4(1) ≟ A4(3), "Segment " + ((((A2(1) * " - ") * A2(3)) * ": ")) + Max, If[Opt ≟ 1 ∧ Max ≟ A4(1) ≟ A4(4), "Segment " + ((((A2(1) * " - ") * A2(4)) * ": ")) + Max, If[Opt ≟ 1 ∧ Max ≟ A4(1) ≟ A4(5), "Segment " + ((((A2(1) * " - ") * A2(5)) * ": ")) + Max, If[Opt ≟ 1 ∧ Max ≟ A4(2) ≟ A4(3), "Segment " + ((((A2(2) * " - ") * A2(3)) * ": ")) + Max, If[Opt ≟ 1 ∧ Max ≟ A4(2) ≟ A4(4), "Segment " + ((((A2(2) * " - ") * A2(4)) * ": ")) + Max, If[Opt ≟ 1 ∧ Max ≟ A4(2) ≟ A4(5), "Segment " + ((((A2(2) * " - ") * A2(5)) * ": ")) + Max, If[Opt ≟ 1 ∧ Max ≟ A4(3) ≟ A4(4), "Segment " + ((((A2(3) * " - ") * A2(4)) * ": ")) + Max, If[Opt ≟ 1 ∧ Max ≟ A4(3) ≟ A4(5), "Segment " + ((((A2(3) * " - ") * A2(5)) * ": ")) + Max, If[Opt ≟ 1 ∧ Max ≟ A4(4) ≟ A4(5), "Segment " + ((((A2(4) * " - ") * A2(5)) * ": ")) + Max, If[Opt ≟ 0 ∧ Min ≟ A4(1) ≟ A4(2), "Segment " + ((((A2(1) * " - ") * A2(2)) * ": ")) + Min, If[Opt ≟ 0 ∧ Min ≟ A4(1) ≟ A4(3), "Segment " + ((((A2(1) * " - ") * A2(3)) * ": ")) + Min, If[Opt ≟ 0 ∧ Min ≟ A4(1) ≟ A4(4), "Segment " + ((((A2(1) * " - ") * A2(4)) * ": ")) + Min, If[Opt ≟ 0 ∧ Max ≟ A4(1) ≟ A4(5), "Segment " + ((((A2(1) * " - ") * A2(5)) * ": ")) + Min, If[Opt ≟ 0 ∧ Min ≟ A4(2) ≟ A4(3), "Segment " + ((((A2(2) * " - ") * A2(3)) * ": ")) + Min, If[Opt ≟ 0 ∧ Min ≟ A4(2) ≟ A4(4), "Segment " + ((((A2(2) * " - ") * A2(4)) * ": ")) + Min, If[Opt ≟ 0 ∧ Min ≟ A4(2) ≟ A4(5), "Segment " + ((((A2(2) * " - ") * A2(5)) * ": ")) + Min, If[Opt ≟ 0 ∧ Min ≟ A4(3) ≟ A4(4), "Segment " + ((((A2(3) * " - ") * A2(4)) * ": ")) + Min, If[Opt ≟ 0 ∧ Min ≟ A4(3) ≟ A4(5), "Segment " + ((((A2(3) * " - ") * A2(5)) * ": ")) + Min, If[Opt ≟ 0 ∧ Min ≟ A4(4) ≟ A4(5), "Segment " + ((((A2(4) * " - ") * A2(5)) * ": ")) + Min, If[Opt ≟ 1 ∧ Max ≟ A4(1), (A2(1) * ": ") + Max, If[Opt ≟ 1 ∧ Max ≟ A4(2), (A2(2) * ": ") + Max, If[Opt ≟ 1 ∧ Max ≟ A4(3), (A2(3) * ": ") + Max, If[Opt ≟ 1 ∧ Max ≟ A4(4), (A2(4) * ": ") + Max, If[Opt ≟ 1 ∧ Max ≟ A4(5), (A2(5) * ": ") + Max, If[Opt ≟ 0 ∧ Min ≟ A4(1), (A2(1) * ": ") + Min, If[Opt ≟ 0 ∧ Min ≟ A4(2), (A2(2) * ": ") + Min, If[Opt ≟ 0 ∧ Min ≟ A4(3), (A2(3) * ": ") + Min, If[Opt ≟ 0 ∧ Min ≟ A4(4), (A2(4) * ": ") + Min, If[Opt ≟ 0 ∧ Min ≟ A4(5), (A2(5) * ": ") + Min]]]]]]]]]]]]]]]]]]]]]]]]]]]]]) → A6

AI Drawing Prompt Reference

Draw a feasible region and optimal solution for a linear programming problem. First, draw the half-planes defined by three linear inequalities to form a polygonal feasible region. Then, identify the vertices of this polygon, labeled A to J. Next, based on the user-input coefficients of the linear objective function, calculate the optimal solution location and find the optimal point within the feasible region. Finally, show the trend of the objective function within the feasible region.

The prompt above was compiled from the original drawing commands and is for reference only.