Simplex Method Linear Equations

Question

minimize

c=2s+t+3u

Subject to

s+u≥100

2s+t≥50

s≥0, t≥0, u≥0

Answer 1

We'll put C in terms of one other variable and either graph it or find a derivative to minimize.

S, T, and U are positive and C is their sum so also positive. Thus S, T, and U will each be as small a positive number as possible, so I'm going to assume S+U=100 and 2S+T=50 for this minimum.

Rearrange: U=100-S and T=50-2S

Substitute: C=2S + (50-2S) + 3(100-S)

Simplify: C=350-3S

Derivative: dC/dS = -3

What this means is that C will decrease by 3 for each increase of S (based on our assumption, at least).

Using the inequalities, maximize S while minimizing the other variables.

S+U≥100 so S=100, U=0, and 2S+T≥50, 200+T≥50, so T=0

Plug into C.

C=200.

Just for some perspective:

s=100, t=0, u=0 >> c=200 (our answer)

s=99, t=0, u=1 >> c=201

s=75, t=0, u=25 >> c=225

s=25, t=0, u=75 >> c=275

s=0, t=50, u=100 >> c=350