Are my constraints in this LP problem correct? Any redundant?

Question

I have to set up an LP problem based on the situation below:

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What I tried:

Let $b_i$ denote sacks bought at month i (i=1,2,3)

Let $s_i$ denote sacks sold at month i (i=2,3,4)

We want to maximise net return (profit?)

$$z = -28.5b_1 + 31s_1 - 30.5b_2 + 32.5s_2 - 29b_3 + 29.5s_3$$

s.t.

Final inventory must be 20,000

1. $$10,000 - \sum_i s_i + \sum_i b_i = 20,000$$

Nonnegative:

2. $$s_i, b_i \ge 0$$

Cash balance:

3. $$\sum_i c_i b_i \le 2,000,000$$

where $c_1 = 28.5, c_2 = 30.5, c_3 = 29$

Capacity of 50,000 sacks:

4. $$10,000 + b_1 \le 50,000$$

5. $$10,000 + b_1 - s_1 + b_2 \le 50,000$$

6. $$10,000 + b_1 - s_1 + b_2 - s_2 + b_3 \le 50,000$$

7. $$10,000 + b_1 - s_1 + b_2 - s_2 + b_3 - s_3 \le 50,000$$

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Is that right?

Is constraint #7 redundant due to constraint #1?

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From Chapter 2 here.

Answer 1

Your third constraint : $$28.5b_1 + 30.5b_2 + 29b_3\le 2,000,000$$ is too restrictive on the cash balance. You don't take into account any revenue in the period.

Try instead: $$28.5b_1 \le 2,000,000$$ $$28.5b_1 - 31s_1 + 30.5b_2 \le 2,000,000$$ $$28.5b_1 - 31s_1 + 30.5b_2 - 32.5s_2 + 29b_3 \le 2,000,000$$

Yes constraint #7 is redundant due to constraint #1 because constraint #1 is more restrictive.