SIMPLEX EXAMPLE
A paint manufacturer produces two types of paint; water based and solvent based. The goal is to maximize the profit by producing right combination of paint types.
Profit margin for water based paint is 50 usd/ton while it is 30 usd/ton for solvent based type.
Material 1 and Material 2 are main raw materials; 6 gallons of Material 1 and 1 gallon of Material 2 are used for a ton of water based paint. 4 gallons of Material 1 and 2 gallons of Material 2 are used for a ton of solvent based paint.
The raw materials have limitation for daily supply; 24 gallons for Material 1 and 6 gallons for Material 2.
Another important criteria is market demand. If the products cannot be sold, the profit maximization will be just on a paper, although the production optimized. According to the sales reports, maximum of 2 tons of water based paint is sold per day. Solvent based paint is sold 1 ton less than water based paint.
Mathematical modelling and simplex solution of this case are as follows;
Maximize Z = 50.00x1 +40.00x2
Constraints:
6.00x1 + 4.00x2 <= 24.00
1.00x1 + 2.00x2 <= 6.00
-1.00x1 + 1.00x2 <= 1.00
Initial Table
Z x1 x2 s1 s2 s3
Z 0.00 50.00 40.00 0.00 0.00 0.00
s1 24.00 6.00 4.00 1.00 0.00 0.00
s2 6.00 1.00 2.00 0.00 1.00 0.00
s3 1.00 -1.00 1.00 0.00 0.00 1.00
Iteration: 1
Incoming vector; x1 (Row 0 Value = 50.00), Outgoing Vector; s1 (Ratio = 4.00), Pivot Value = 6.00
New Table
Z x1 x2 s1 s2 s3
Z -200.00 0.00 6.67 -8.33 0.00 0.00
x1 4.00 1.00 0.67 0.17 0.00 0.00
s2 2.00 0.00 1.33 -0.17 1.00 0.00
s3 5.00 0.00 1.67 0.17 0.00 1.00
Iteration: 2
Incoming vector; x2 (Row 0 Value = 6.67), Outgoing Vector; s2 (Ratio = 1.50), Pivot Value = 1.33
New Table
Z x1 x2 s1 s2 s3
Z -210.00 0.00 0.00 -7.50 -5.00 0.00
x1 3.00 1.00 0.00 0.25 -0.50 0.00
x2 1.50 0.00 1.00 -0.12 0.75 0.00
s3 2.50 0.00 0.00 0.37 -1.25 1.00
The problem has optimum solution
***** RESULTS: *****
Zmax = 210.0000000000
x1 = 3.0000000000
x2 = 1.5000000000
Elapsed Time for Calculation: 0.03200000 seconds
