simplex algorithm - minimization

Question

So I get the basic concept of simplex algorithm but I am working on a project where I have to implement any linear programming algorithm (I chose simplex method) to minimize a function, but I don't have enough numbers (stay with me here). Basically, I tried by setting M_fe = 0 and M_sp = 0, but I still need a value for sigma in order to get the values of the 8 variables in eqn(1) and 3 variables in eqn(2). Variables in eqn(1) and eqn(2) hold the same value. Here's the following information:

Constraint equation:

4.534*F_bic + 3.167*F_bra + 2.930*F_brd + 0.174*F_pro - 1.997*F_tri + 0.02*F_cr + 0.248*F_ecu + 0.726*F_ecrl = M_fe (moment of flexion) ----------------- eqn(1)

1.28*F_bic - 0.1954*F_brd - 0.1736*F_pro = M_sp (moment of supination) ---- eqn(2)

Inequality constraint:

F_i/PCSA_i <= sigma ----------------- eqn(3)

Objective function:

Minimizing sigma ---------- eqn(4)

Some conditions: M_fe can be equal to zero if that makes the problem easier to solve. PCSA_i are variables and their values are given as follows:

1. PCSA_bic = 4.6
2. PCSA_bra = 7.0
3. PCSA_brd = 1.5
4. PCSA_pro = 3.4
5. PCSA_tri = 18.8
6. PCSA_fcr = 2.0
7. PCSA_ecu = 3.4
8. PCSA_ecrl = 2.4

Basically, my question is that is there any way to solve this problem without having a value for sigma? Funny thing is sigma is the same in eqn(3) for all 8 variables, which made me plugin the value of sigma for each corresponding variable in eqn(1) and eqn(2). Doing that gave me

13.094*sigma = M_fe ---------- eqn(5)

5.005*sigma = M_sp ----------- eqn(6)

But now how would i solve for M_fe and M_sp. This problem is like a merry-go-round. Any help is greatly appreciated. Also, this is my first post and technically my first time so please forgive me if I didn't follow any specific protocols in this post.

Thanks.

Answer 1

So I actually figured out how to solve this problem in May but never had a chance to post my answer. So sigma is minimized and the inequality constraints can then be solved. Using a simplex algorithm, we can set the equation as follows:

Ax = B

Aeq = [4.534 3.167 2.930 0.248 0.0201 0.716]

Beq = [20]

x = [FBIC; FBRA; FBRD; FECU; FFCR; FECRL]

The values from x can then easily give us the F_i value, which I was trying to solve for in equation 3.

Although this problem was very specific and extremely difficult, similar simpler problems can also be solved using a simplex algorithm and the following equation set up.

I can also share my MATLAB code if anyone is further interested in looking at this problem.

Cheers!