Linear Programming using the simplex method

In summary, 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=$22 There is also a shipping tax per barrel: S1: D1=$2, D2=$6 S2: D1=$5, D2=$4 Southwestern Oil is determined to spend no more than $40,000 on
  • #1
arl2267
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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.
 
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  • #2
arl2267 said:
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.
 

FAQ: Linear Programming using the simplex method

1. What is linear programming?

Linear programming is a mathematical method used to find the best solution to a problem with linear constraints. It involves optimizing an objective function while staying within the limits set by certain constraints.

2. What is the simplex method?

The simplex method is an algorithm used to solve linear programming problems. It involves systematically moving from one feasible solution to another until the optimal solution is found.

3. How does the simplex method work?

The simplex method works by starting at a feasible solution and moving to adjacent feasible solutions until the optimal solution is reached. It uses a set of rules to determine which adjacent solution to move to, and repeats this process until no further improvement can be made.

4. What are the steps involved in the simplex method?

The steps involved in the simplex method are:

  1. Formulating the problem in standard form
  2. Creating the initial simplex tableau
  3. Selecting the pivot column and row
  4. Performing row operations to make the pivot element 1 and all other elements in the pivot column 0
  5. Repeating the pivot process until the optimal solution is reached

5. What are the advantages of using the simplex method?

The advantages of using the simplex method include:

  • It is a well-established and widely used method for solving linear programming problems
  • It is relatively easy to understand and implement
  • It guarantees finding the optimal solution, if one exists
  • It can handle large, complex problems with many variables and constraints

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