Linear Programming using the simplex method

[arl2267]

Southwestern Oil supplies two distributors in the Northwest from two outlets. S1 and S2. Distributor S1 needs at least 3000 barrels of oil, and D2 needs at least 5000 barrels. The two outlets can each furnish exactly 5000 barrels of oil. The cost per barrel to ship the oil are:S1: D1=$30, D2=$20
S2: D1=$25, D2=$22There is also a shipping tax per barrel:S1: D1=$2, D2=$6
S2: D1=$5, D2=$4Southwestern Oil is determined to spend no more than $40,000 on shipping tax.a) How should the oil be supplied to minimize cost?
b) Find and interpret the values of any nonzero slack or surplus variableOkay so my attempt at coming up with the constraints is this:

Minimum: W=
30x₁+20x₂>=3000
25x₃+22x₄>=5000
x₁+x₂=50,000I think what is throwing me off is the shipping tax. I understand that the forum rules are that we need to make an attempt, but I am having such a hard time with this, and would really appreciate some help.

 

[Sudharaka]

Southwestern Oil supplies two distributors in the Northwest from two outlets. S1 and S2. Distributor D1 needs at least 3000 barrels of oil, and D2 needs at least 5000 barrels. The two outlets can each furnish exactly 5000 barrels of oil. The cost per barrel to ship the oil are:S1: D1=$30, D2=$20
S2: D1=$25, D2=$22There is also a shipping tax per barrel:S1: D1=$2, D2=$6
S2: D1=$5, D2=$4Southwestern Oil is determined to spend no more than $40,000 on shipping tax.a) How should the oil be supplied to minimize cost?
b) Find and interpret the values of any nonzero slack or surplus variableOkay so my attempt at coming up with the constraints is this:

Minimum: W=
30x₁+20x₂>=3000
25x₃+22x₄>=5000
x₁+x₂=50,000I think what is throwing me off is the shipping tax. I understand that the forum rules are that we need to make an attempt, but I am having such a hard time with this, and would really appreciate some help.

Hi arl2267, :)

Welcome to Math Help Boards! :)

First define your variables as,

\(x_{ij}\) - The number for barrels supplied from \(S_{i}\) to distributor \(D_{j}\) where \(i,j=1,2\)

So the total cost will be, \(z=(30+2)x_{11}+(20+6)x_{12}+(25+5)x_{21}+(22+4)x_{22}\). Hence the objective function is,

\[\mbox{Min }z=32x_{11}+26x_{12}+30x_{21}+26x_{22}\]

Since \(D1\) needs at least \(3000\) barrels of oil we have,

\[x_{11}+x_{21}\geq 3000\]

Can you try to obtain the rest of the constraints? :)

Kind Regards,
Sudharaka.