1. Constrained Optimization
Chapter 15
Constrained Optimization

2. The Linear Programming Model
Let: x1, x2, x3, ………, xn = decision variables
Z = Objective function or linear function
Objective: Maximize Z = c1x1 + c2x2 + c3x3 + ………+ cnxn
subject to the following constraints:
where aij, bi, and cj are given constants.

3. The Linear Programming Model
In more efficient notation:
The decision variables, xI, x2, ..., xn, represent levels of n competing activities.

4. EXAMPLE: Gas Processing Problem
(Chemical/Petroleum Engineering Problem)
Company receives a fixed amount of raw gas each week.
Processes it to produce two grades of heating gas, regular and premium quality.
Each yields different profits, and involves different time and on-site storage constraints.
Objective: maximize profit without violating the material, time, and storage constraints
Solve using:
Graphical Method
Simplex Method
Excel solver (Simplex LP)

5. Graphical Solution

7. Aside from a single optimal solution; there are three other possible outcomes:
Alternative optima
no feasible solution
an unbounded result

8. The Simplex Method
When decision variables are more than 2, it is always advisable to use Simplex Method to avoid lengthy graphical procedure.
It does not examine all the feasible solutions. Only the extreme points
It deals only with a small and unique set of feasible solutions, the set of vertex points (i.e., extreme points) of the convex feasible space that contains the optimal solution.

9. The Simplex Method Steps involved:
Locate an extreme point of the feasible region.
Examine each boundary edge intersecting at this point to see whether movement along any edge increases the value of the objective function.
If the value of the objective function increases along any edge, move along this edge to the adjacent extreme point. If several edges indicate improvement, the edge providing the greatest rate of increase is selected.
Repeat steps 2 and 3 until movement along any edge no longer increases the value of the objective function.

10. **Here

11. Try point A (x1=0, x2=0) first.
Then pick a leaving variable and an entering variable which will make Z bigger.
In this example, x2 should be chosen to enter but to make it simple we will choose x1 to enter. S2 will become a nonbasic variable (the leaving variable).
This will take us to point B.

12. Excel Solution Gas Processing Problem
*Write down the equations on the Excel Sheet.
*Demonstrate how to use the solver in Excel (DATA  Solver) (fileoptionsadd-ins)