Dual problem and primal problem unbounded linear programming

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Klaus Hajdaraj 2021-4-14

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https://ww2.mathworks.cn/matlabcentral/answers/801986-dual-problem-and-primal-problem-unbounded-linear-programming

Hello,

I need some help please to solve this problem I have . I have solved a max. problem , the primal problem , by using linprog.

After this , I have done the dual problem of the primal. After this proccess, I don't get the result that I should get , because the values of F0 should be the same in both cases(if I am not wrong).

I am sending the illustrated problem and code below. I hope someone can give me a help.

thank you !

Max 300𝑥1 + 400𝑥2

𝑥1 + 𝑥2 ≤ 320

4𝑥1 + 5𝑥2 ≤ 510

2𝑥1 + 3𝑥2 ≤ 430

3𝑥1 + 𝑥2 ≤ 300

𝑥1,𝑥2 ≥ 0

The Dual Problem

Min 320𝑦1 + 510𝑦2 + 430𝑦3 + 300𝑦4

𝑦1 + 4𝑦2 + 2𝑦3 + 3𝑦4 ≥ 300

4𝑦1 + 5𝑦2 + 3𝑦3 + 𝑦4 ≥ 400

𝑦1,𝑦2,𝑦3,𝑦4 ≥ 0

Matlab Code:

%Primary Problem
%Ax<=b
%coefficients of A
A = [1 4; 4 5; 2 3; 3 1];
%coefficients of B
b = [320 510 430 300];
%coefficients of the objective function
f = [300 400];
%The maximation of the linear function using the matlab linear 
%programming function 
[x0,F0] = linprog(-f,A,b,[],[],[0 0])
% disp('x0 = ')
% disp(x0);
%------------------------------------------------------------------
%Dual problem
%coefficients of AT
AT = [1 4 2 3;
      4 5 3 1];
%coefficients of u
u = [300 400];
g = [320 510 430 300];
%setting a lower bounder
lb = zeros(1,4);
%setting an upper bounder
ub=[];
[u0,F0] = linprog(g,-AT,-u,[],[],lb,ub)
% disp('u0 = ');
% disp(u0);
%So,1st and 2nd constrains of the primary problem are active
%because u0 = [9.0909
%              72.7273
%              0
%              0]
%Solutions of Primary problem throug the dual problem
syms x1 x2
eqns = [x1 + x2 == 320, 4*x1 + 5*x2 == 510];
S = solve(eqns,[x1 x2]);
b = [S.x1 S.x2];
disp('Values of x ')
disp(b);
duality_gap = [x0(1)-u0(1); x0(2)-u0(2)]
            