# Problem with the simplex method in a cost accounting problem

I have the following problem in my business class. I have done an error in my solutions but I don't know where it is. Can Someone help me?

It's October, and the new management of Elektronik AG is determined to achieve its climate goals by the end of the year in line with the Paris Climate Agreement. To achieve this, they have enlisted the expertise of a specialist to estimate the levels of greenhouse gas emissions and water consumption resulting from the production of their electronic products. The individual values can be found in the table below. In order to meet the year-end goals, the company must limit its greenhouse gas emissions to only 1500 metric tons and its water consumption to 60 million liters. Therefore, the management is requesting that you propose an optimized production program that takes into account the climate goals while maximizing the highest possible contribution margin.

a) Which products should be considered for the optimization of the production program using the Simplex algorithm? Justify your selection based on relative contribution margins.

b) Now, taking into consideration the results from a), formally articulate the optimization problem. (6 points)

c) In preparation for determining the optimal production program using the Simplex algorithm, set up the initial tableau. Also, identify the pivot column and row, and the pivot element. Justify your selection.

Now, for part a), I have calculated the relative contribution margins, and they should be as follows:"

While for part b), I have formulated the following optimization problem.

$$\max CB(x_l,x_t,x_s)=1000x_l+300x_t+210x_s \\ 0 \leq x_l\leq 2000 \\ 0 \leq x_t\leq 5000 \\ 0 \leq x_s\leq 10000 \\ 4x_l+2.5x_t+3x_s \leq 1500000 \\ 0.05x_l+0.1x_t+0.21 x_s\leq 60000000$$

Now I believe there may be an issue with my formulation, as in part c) (which I've verified using an online calculator), it seems that all three products can be produced without any constraints, which doesn't make sense.

Can you tell me where I'm going wrong?

\begin{align} \max z = 1,000x_l+300x_t+210x_s\\ 0 \leq x_l \leq 2000 \\ 0 \leq x_t \leq 5000 \\ 0 \leq x_s \leq 10,000 \\ 250 x_s + 120x_t + 70x_s \leq 1,500,000 \\ 20,000 x_s + 3,000x_t + 1,000x_s \leq 60,000,000 \\ \end{align}