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### Transcript of NON-LINEAR PROGRAMMING (NLP) ¢â‚¬¢ Linear programming, integer programming...

• PROF. DR. VEDAT CEYHAN

NON-LINEAR PROGRAMMING (NLP)

• Similarities & differences

Characteristics of NLP

NLP models

One Variable NLP

Multi Variable NLP

Unlimited NLP

Limited NLP

 Basic concept

 Case studies

2

• Similarities & differences

• Linear programming(LP) – Linear target function+ linear constraints

– Continuous variable

• Integer programming (IP) – Linear target function+ linear constraints

– Discrete variable

• Non-linear programming (NLP) – The objective is linear but constraints are non-linear

– Non-linear objectives and linear constraints

– Non-linear objectives and constraints

3

• Characteristics of NLP

Solution is difficult

Solution may tie initial point.

Initial point is subjective.

4

• • NLP forms: – Non-linear objectives

– Non-linear constraints

– Non-linear objectives and constraints

The main problem is algorithm selection

2min ( -3)

. . x< 3

x

s t

1 2

2 2 1 2

min

. . 4

x x

s t x x

 

2 1 2

2 2 1 2

min ( 2)

. . 4

x x

s t x x

 

 

5

• • Constructing model is required for solving optimization problem. Mathematical model is based on determination of variables and defining their functional relationship

• 3 components of mathematical model

– Objective function,

– Constraints

– Non-negativity restriction

6

• • Linear programming, integer programming and goal programming assume that objective function and constraints is linear. Not include non-linear expressions such as 1/X2, log X3

• NLP procedures does not produce optimum solution every time in contrary with linear programming (Render ve ark, 2012).

7

• Objective function in NLP

• gi(x) ≤ bi constraints

• z = f(x) objective function

We find the vector of x= (x1, x2,…, xn), which produce optimum solution for objective

8

• NLP models

• Associated with the number of variable

–One variable or multivariate,

 Associated with the presence of restrictions

– restricted or unrestricted

9

• NLP models

Constraints: equality

Limited Model

One variable models Constraints: inequality

Unrestricted Model

Constraints: equality

Limited Model

Multivariate models Constraints:inequality Unresticted Model

10

• Basic concepts

• Increasing and decreasing function

y=f(x)

x1 and x2 is a random figüre

If the function is f(x1)

• Increasing and decreasing function

Increasing function Decreasing function

12

• – Local Maximum and Minimum

f(x)’’> f(x)’

f(x)’’< f(x)’ local min. or max

Gradiant function of Z=f(x1,x2,…,xn) is vector of first derivatives.

13

• Hessian Matrix

• Hessian matrix is a nxn matrix of second partial derivative of f(x1,x2,…,xn) function

14

• In LP, solution region is convex set and optimum solution is one of the corner point

In NLP, it is not necessary that optimum solution is one of the corner point

15

• Concave function Convex function

16

• Example:

1) Is f(x)= √x function whether convex or concave at [S= 0, ∞)?

f(x)=√x concave function

Since line between two point take place under the curve, the

function is concave. 17

• 2) Is f(x)=x3 function whether convex or concave at S = R1=(-∞, ∞)?

f(x)=x³ Neither convex nor concave function

18

• 3) Is f(x)=3x-3 function whether convex or concave at S = R1=(-∞, ∞)?

1

Both convex and concave function

-3

19

• One variable unrstricted NLP

• Deal with finding maximum or minimum point of one variable function with no restriction

20

• Multivariate unrestricted NLP

• Deal with finding optimum point (x1*,x2*,…,xn*) that is maximum or minimum point of multivariate function of f(x1,x2,…,xn) with no restriction

• Demand function explained by price, preference, income, etc. İs a example for multivariate unrestricted NLP.

21

• Restricted NLP

• Examining the colineairty among x1,x2,……,xn that is variables of multivariate function of f(x1,x2,……,xn

• f(x1,x2,……,xn) function,

• g1(x1,x2,……,xn) = b1 • g2(x1,x2,……,xn) =b2 restrictions

• gm(x1,x2,……,xn) =bm

We are looking for point that is maximum or minimum under upper restrictions.

22

• Solution

• Lagrange function

L(x1,x2,…………….xn,λ1,λ2,……….λm)= f(x1,x2,……..xn) + i[ bi- gi (x1,x2,………xn)]

23

• Example

1 1

2

2 2

1 2

1 2

1 2

1 2

2 2

1 2 1 1 2 2 1 2

1 1 2 1

2

Minimize ( 4) ( 4)

subject to 2 3 6

3 2 12

and , 0

The Lagrangian is:

( 4) ( 4) (6 2 3 ) ( 12 3 2 )

Kuhn Tucker Conditions:

2( 4) 2 3 0 0 and 0

2( 4)

x x

x

C x x

x x

x x

x x

Z x x x x x x

Z x x x Z

Z x

 

 

   

 

   

          

      

  2

1 1

2 2

1 2 2

1 2 1 1

1 2 2 2

28 36 16 1 2 1 213 13 13

3 2 0 0 and 0

6 2 3 0 0 and 0

12 3 2 0 0 and 0

Solution: , , 0,

xy x Z

Z x x Z

Z x x Z

x x

 

 

 

 

 

 

    

     

      

   

24

• Non-linear objective function and linear constraints

25

• Case of non-linear objective function and linear constraints

 Great Western Appliance firm sell toast machine of Mikrotoaster (X1) and Self-Clean Toaster Oven (X2). Firm gain net profit by \$28 per toaster. Profit function is 21X2+0,25

 Objective function is non-linear:

Maximum profit =28X1+21X2+0,25

 There are two linear constraints

X1+X2 ≤ 1,000 (production capacity)

0.5X1+0.4X2 ≤ 500 (term of sale)

X1, X2 ≥ 0

(Source: Render ve ark., 2012)

26

Objective function includes terms such as 0,25 and when the restrictions is linear

(Source: Render and ark., 2012)

27

• EXAMPLES

28

• WIN QSB for NLP

29

• WİN QSB interface

30

• WİN QSB EKRANINI TANIYALIM

31

• Unrestricted NLP example

• minimum x(sin(3.14159x))

• 0

• 33

For unrestricted case, enter

“0”

• Unrestricted NLP example

34

• 35

Attention!

Interval of X1

Graphical solution

• x=3.5287 Objective function= -3.5144.

36

• Example 2:

Objective function:

maximum 2x1 + x2 - 5loge(x1)sin(x2)

Constraints x1x2

• 38

• 39

• x1=3.3340 ve x2=2.9997 Objective function=8.8166

40

• Non-linear function and constraints

41

• • In medium size hospital having 200-400 patient bed, Hospicare Corporation, annual profit depend on number of patient(X1) ve number of patient surgeon(X2) bağlıdır.

• Non-linear objective function for Hospicare : \$13X1 + \$6X1X2 + \$5X2 + \$1/X2

Constraints;

• 2X1 2+4X2≤90 (nursery capacity)

• X1+X2 3≤75 (X-ray capacity)

• 8X1-2X2≤61 (marketing budget)

42

• 43

• 44

• 45

• Example: Pickens Memorial Hospital

Patient demand exceeds capacity of hospital

Objective: Maximum profit

46

• Decision variables

M =number of served patient

S = number of served patient for surgery

P = number of served child patient

Profit function When increasing patient, profit increasing non-

linearly.

47

• Constraints

• Hospital capacity: Total 200 patient

• X-ray capacity: 560 x-rays per week

• Marketing budget: \$100