Chapter 15

Constrained Optimization

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.

• In more efficient notation:

The decision variables, xI, x2, ..., xn, represent levels of n competing activities.

The Linear Programming Model

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:1.Graphical Method 2.Simplex Method3.Excel solver (Simplex LP)

s)Constraint (Positive 0,

)Constraint Storage Premium"(" 6

)Constraint Storage Regular"(" 9

)Constraint (Time 80810 : workedhours Total

)Constraint (Material 77117 :used gas raw Total

175150 :Profit Maximize

:gas Premium :gasRegular

:nFormulatio gProgramminLinear

21

2

1

21

21

21

21

xx

x

x

xx

xx

xxZ

xx

Graphical Solution

s)Constraint (Positive 0, 6) (5,

)Constraint Storage Premium"(" 6 (4)

)Constraint Storage Regular"(" 9 (3)

)Constraint (Time 80810 (2)

)Constraint (Material 77117 (1)

175150

:gas Premium :gasRegular

21

2

1

21

21

21

21

xx

x

x

xx

xx

xxZ

xx

Aside from a single optimal solution; there are three other possible outcomes:(a) Alternative optima(b) no feasible solution(c) an unbounded result

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.

The Simplex Method

Steps involved:

1. Locate an extreme point of the feasible region.

2. Examine each boundary edge intersecting at this point to see whether movement along any edge increases the value of the objective function.

3. 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.

4. Repeat steps 2 and 3 until movement along any edge no longer increases the value of the objective function.

The Simplex Method

1 2

1 2

1 2 1

1 2

Regular gas: Premium gas:

150 175

Introduce slack variables; then inequalities become equalities

(1) 7 11 77

(2) 10 8

x x

Z x x

x x S

x x

2

1 3

2 4

1 2 1 2 3 4

80

(3) 9

(4) 6

, , , , , 0 (Positive Co

S

x S

x S

x x S S S S

nstraints)

4 Equations, 6 unknowns

Set 2 of them to zero each time (nonbasic variables)

and solve the resultant set of equations.

**Here

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.

Gas Processing Problem*Write down the equations on the Excel Sheet.*Demonstrate how to use the solver in Excel

Excel Solution