Simplex method: initial tableau from a given feasible solution I have a linear program that I want to solve using the Simplex Method. I would like help figuring out how to initialize the tableau from a given feasible solution. 
I have found many instructions how to initialize the tableau generally. But they all involve introducing slack/artificial variables in order to derive an initial feasible solution where all the original variables are set to $0$. However, for my program I already know an initial solution from which I would like to start.
Concretely my program is
$$\text{max } -\theta \text{ subject to}$$
$$(\sum_{i = 1}^{n} c_i^{(1)}\lambda_i)  - c_k^{(1)}\theta + s^{(1)} = 0$$
$$(\sum_{i = 1}^{n} c_i^{(2)}\lambda_i)  - c_k^{(2)}\theta + s^{(2)} = 0$$
$$(\sum_{i = 1}^{n} c_i^{(3)}\lambda_i)  - c_k^{(3)}\theta + s^{(3)} = 0$$
$$(\sum_{i = 1}^{n} c_i^{(4)}\lambda_i)  - s^{(4)} = c_k^{(4)}$$
$$\sum_{i = 1}^{n} \lambda_i = 1$$ 
where $k$ is some number in $\{1, \ldots, n\}$ and the $c_i^{(j)}$'s are nonnegative constants. In this case setting $\theta = \lambda_k = 1$ and all other variables to $0$ would be a solution. How would I construct the initial tableau given this initial solution?
 A: In any case you MUST introduce artificial variables. There is no way to avoid them. I mean you started from some Corner Point, okay. But then? To walk through other points you need to introduce slack variables, actually using M method — google it. As all your constraints are not in general form, then you should add slack variables with $M$ coefficient, which if very big number, in all of them.
And if we assume that you have 5 constraints in your problem, then you will have 5 slack variables. Then, as I understand, if you know your initial feasible solution without introducing slack variables, then your slack variables will be of value $0$. And others are those you have as initial values. It means you make your slack variables be non-basic, and others are basic. Hope this helps.
P.S.: look at this answer too. https://stackoverflow.com/questions/17289032/solving-a-linear-program-in-case-of-an-equality-constraint
As mentioned there you can also change equalities to 2 inequalities, but by doing so you also increase the number of constraints. So you can choose either M-method or replacing equalities with inequalities.
