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

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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 of function

    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

  • 2. Quadratic programming

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

    Quadratic programming problems can be solved by using adjusted simplex algorithms.

    (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)

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  • 43

  • 44

  • 45

  • Example: Pickens Memorial Hospital

    Patient demand exceeds capacity of hospital

    Objective: Maximum profit

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  • 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