OPTIMIZATION TECHIQUE vaious method simplex

1. Engineering Optimization
Techniques
Presented By
ABC
PhDME

2. Problem
• The are different dumping sites in surrounding area of Islamabad/Rawalpindi for municipal waste, Out
of all of them four sites named as
• Chohan Chakri Dump Site,
• National Park dump site, Taxila,
• H-11 dump sites
• Fateh Jung dump site
were selected for collection of municipal waste for shifting/trasnsporting of municipal waste to two
power plants (operated from municipal waste) for power generation located in Rawat and Fateh Junk.
• The objective of this project is to minimize the cost of transportation

3. Solution
• The first thing in this problem is the cost of transportation from dumping sites to plants which is given
(in Thousands) in table below
• Cost is estimated as per distance given in google maps
Dump Sites Rawat Fateh Jung
Chakri 3.377 4.64
Taxila 3.04 4.06
H-11 2.17 5.02
Fateh Jung 1.06 6.09

4. Solution
• The first thing in this problem is the available municipal
waste at various dumping sites given in table below
Dump Sites Amound of waste
on Tonnes
Chakri 100
Taxila 150
H-11 250
Fateh Jung 90

5. Q is the internal heat source (heat generated per unit time per unit volume
is positive),
Solution
• The third thing in this problem is the requirement of municipal waste at various dumping sites
given in table below
Requirement of waste in Tones
Plant 1 Plant 2
200 350

6. Solution
• Figure shows the transportation of wastes from dumping
sites. Where x1, x2 x3 are the total cost of shiping, which
will determined to minimize the total cost

7. Function
• Minimizing function is given by
f = 3.77x1 + 3.045x3 + 2.17x5 + 1.06x7 + 4.64 x2 + 4.06x4 + 5.02x6 + 6.09x8
• Subjected to constraints
• X1 + X5 ≤ 100
• X2 + X6 ≤ 150
• X3 + X7 ≤ 250
• X4 + X8 ≤ 250
• X1 + X2 + X3 +X4 ≥ 200
• X5 + X6 + X7 +X8 ≥ 350
• x1, x2, x3, 4, x5, x6, X7, X8 ≥ 0.

8. Solution by simplex Method

9. Solution by simplex Method (Coefficients
and constraints)

10. Solution by simplex Method

11. Solution by simplex Method (Solver)

12. Solution by simplex Method (Solver)
Minimized Cost in thousands is 1909.1

13. Solution by simplex Method
Minimized Cost in thousands is 1909.1

14. Solution by simplex Method

15. Modification 1
• The problem is now modified using different objective coefficients.
(Transportation cost is not same for all transporting vehicles)
Minimized Cost in thousands is 1567.1

16. Solution by simplex Method

17. Modification 2
100 250

18. Solution by simplex Method