Consider the linear programming problem: minimize z = 4x_1 + x_2 subject to: 0.2
ID: 3221355 • Letter: C
Question
Consider the linear programming problem: minimize z = 4x_1 + x_2 subject to: 0.25 x_1 + x_2 greaterthanorequalto 2 7x_1 + 4x_2 greaterthanorequalto 32 x_1 greaterthanorequalto 0, x_2 greaterthanorequalto 0 Obtain the solution: i. graphically; ii. using the simplex method; iii. using an appropriate software package (e.g.. Excel, MATLAB, etc.). (b) The tableaus shown below were obtained in the course of solving linear programs (where the objective function z is maximized) with 2 non-negative variables (X_1 and x_2) and 2 inequality constraints (i.e., corresponding to the two slack variables S_1 and S_2). In each case, indicate whether the linear program: is unbounded; has a unique optimum; has an alternative optimum solution; is degenerate (in this case, indicate whether any of the above (l)-(3) holds); Justify your answers.Explanation / Answer
If we solve the given problem
(i) Graphically, below is the solution
Vertex Lines Through Vertex Value of Objective
(4,1) 0.25x+y = 2; 7x+4y = 32 17
(8,0) 0.25x+y = 2; y = 0 32
(0,8) 7x+4y = 32; x = 0 8 Minimum
ii) simplex method
Tableau #1
x y s1 s2 -z
0.25 1 -1 0 0 2
7 4 0 -1 0 32
4 1 0 0 1 0
Tableau #2
x y s1 s2 -z
0.25 1 -1 0 0 2
6 0 4 -1 0 24
3.75 0 1 0 1 -2
Tableau #3
x y s1 s2 -z
0 1 -1.167 0.0417 0 1
1 0 0.667 -0.167 0 4
0 0 -1.5 0.625 1 -17
Tableau #4
x y s1 s2 -z
1.75 1 0 -0.25 0 8
1.5 0 1 -0.25 0 6
2.25 0 0 0.25 1 -8
Optimal Solution: z = 8 for x = 0, y = 8
iii) Excel Solver
Solver Solution x y Total Demand Decision Variables 0 8 Minimize 4 1 8 Constraint 1 0.25 1 8 2 Constraint 2 7 4 32 32