Goal Programming

1
Goal Programming
Presented by Saman Halgamuge, University of
Melbourne Material and tutorial support Sunil
Adhikari
2
Introduction

• Previously, we dealt with problems with only one
  objective.
• When there are multiple (sometimes conflicting)
  objectives,
• LP cant find a solution to satisfy all the
  goals
• Goal programming seeks a compromised solution
  based on the relative importance of each
  objective,
• In goal programming,
• Variables may be negative
• Constraints may not be strictly binding
• There are two or more objectives

3
Consider this example,

• A mini company produces two items A and B
• Number of working hours per day is 8 hrs.
• Time required to finish product A and B are 20
  min and 40 min respectively
• Goals are to
• Produce at least 20 products for A and at least
  10 products from B

Let
According to the given information
4
Check the feasible region

• There is no feasible region for the problem,
  i.e., all the goals can not be satisfied at the
  same time
• Obtain a decision (solution) according to the
  relative importance of goals
• For example if the loss from not producing 20
  products of A is greater than the loss from not
  producing 10 products of B, then production of
  20 of A is the most important goal.
• This is where goal programming comes into
  action.

5

• The strategy in goal programming is to convert
  the problem into a LP problem using the following
  procedure.
• 1.The goal constraints are converted into
  equality constraints using additional variables
  called deviational variables.
• Deviational Variables represent overachieving or
  underachieving the desired level of each goal
• d Represents overachieving level of the goal
• d- Represents underachieving level of the goal
• 2. Objective function is formulated so as to
  minimize the penalty from violating goals.

e.g.
Decision Variables
Desired Goal Level
- d d-
6
Components

• Economic Constraints (cannot be violated)
• Resource constraints
• Example of production hours for the day

7
Components

• Goal Constraints
• May not be strict
• Concerned with targets to be achieved
• Can be changed/modified/compromised
• Example Desire to achieve a certain level of
  profit

8
Components

• Objective Function
• Minimizes the sum of the weighted deviations from
  the target values this is ALWAYS the objective
  for Goal Programming
• Not the same as LP (which is to maximize
  revenue/minimize costs)

9
Goal Programming Steps

• Define decision variables
• Define Deviational Variable for each goal
• Formulate Constraint Equations
• Economic constraints
• Goal constraints
• Formulate Objective Function

10
Example 1

• A mini company produces two items A and B
• Number of working hours per day is 10 hrs.
• Time required to finish product A and B are 20
  min and 40 min respectively
• Goals are to Produce
• at least 20 products of A
• At least 10 products of B
• Over producing one product will not be penalized

11

• Step1 define the decision variables
• Step2define the goals
• Goal 1 produce at least 20 from A
• Goal 2 produce at least 10 from B
• Step3 define the deviational variables

No of A items produced in excess of the goal of
20
No of A items produced less than the goal of 20
No of B items produced in excess of the goal of
10
No of B items produced less than the goal of 10
12

• Now formulate LP model of the GP problem
• constraints

Goal constraints
Economic constraints

• objective function It is required to produce
• at least 20 items of A
• at least 10 items of B
• Therefore, minimize the total of item A produced
  less than 20 and item B less than 10.

Un-weighted Equal relative importance of goals
When goals have relative importance,
Weighted according to relative importance of goals
13
Graphical solution

• NOTE Select (a,b) as the optimum point. Show
  that B (a,b)
• 1) If a ? 20 and b ? 10
• MIN20-a10-b
• MAX ab
• 2a4b ? 60
• 4ab ? 60 2a
• gt Maxa
• a20 gt B(a,b)
• 2) IF agt20 and b ? 10
• MIN10-b gtpoint B
• 3) If a ? 20 and bgt10
• MIN20-a gt point A

Goal 2
X2
20
15
2X1 4X2 60
Goal 1
10
A
(10,10)
5
B
(20,5)
X1
0
5
10
15
20
25
30
35
14

• At point A

• At point B

• Requirement is to minimize
• So, point B is the optimal point

Interpretation No of products of A 20,
goal 1 achieved No of products of B 5,
goal 2 not achieved
15

• Consider the case when the weighted goals are
  presented
• e.g. in the earlier example, the loss by not
  producing 20 items of A is twice as much as not
  producing 10 items of B
• Then, the objective function would be

16
Preemptive Goal programming algorithms

• In our examples, the exact relative importance of
  goals had been provided.
• But, the decision maker may not always be able to
  find it.
• In that case, preemptive goal programming is a
  useful tool
• A priority order for all the goals should be
  established
• Satisfy the highest priority goal first and try
  to go down the priority list

17
Non linear programming
18
Introduction

• Linear Programming (LP)
• Linear objective function linear constraints
  (all)
• Continuous variables
• Integer Programming (IP)
• Linear objective function linear constraints
  (all)
• Discrete variables
• optimization problems with non-linear objective
  function or non-linear constraints use non-linear
  programming (NLP)

19

• the problem can include the following forms
• Objective function can be non-linear
• Constraints can be non-linear
• Or, non-linear objective and constraints
  functions
• Identifying the type of the problem is important
  when it comes to selecting a proper algorithm.
  Some algorithms can perform faster for some type
  of problems

20
NLP can be defined as
Objective function
Decision variable
Feasible region of the problem
X can be

Unconstrained optimization problem
Unequal constraints

Equal constraints
Bounds
Constrained optimization problem
21

• The point is said to be global minimum of
  f(x) on X, if
• The point is said to be local minimum of
  f(x) on X, if

local optimal points
Global optimal point
22
Unconstrained optimization

• First order necessary condition for a local
  extremum
• The condition does not guarantee
  that x is a local minimum or maximum. It gives
  only stationary points.
• Go to the second order condition to check it
• Second order condition for a local optimum

Suppose that the is differentiable
at x. if x is a local extremum, then
Suppose that is twice differentiable
at x and . If xis a local minimum,
then the determinant of Hessian matrix
(Hessian) is positive semi
definite. If x is a local maximum then the
Hessian is negative semi definite.
If H(x) 0, then x is a saddle
point
23

• e.g consider the single dimensional problem
• For the stationary points from
• 1st order condition
• Then, from the 2nd order condition

Hessian matrix H(x)
24

• Consider the following multi dimensional problem
• There is a stationary point at x12, x22
• Find the Hessian of the function
• Therefore, its a minimum point

25

• For many real world problems, finding an
  analytical solution is difficult
• In that case, iterative methods can be used to
  obtain fast, fairly approximate solutions
• Gradient descent/ascent
• Steepest descent/ascent
• Newtons method

26
Descent methods

• The main concept is
• Start from a one feasible point
• Gradually move towards optimal point with a
  proper searching direction and a step size
• Different algorithms depending on the selected
  searching direction
• Step size determines the rate of convergence

27
Gradient descent method

• Given a starting point
• Repeat
• (search direction
  negative gradient of the function)
• Choose a step size t
• Update
• Until the stopping criteria is satisfied

Algorithm will stop when gradient goes close to
zero
28

• gradient based search methods are easily stuck in
  local optimum, when applying to functions with
  several local points, the For example for an
  objective function like ,
• 
  which has several local points.
• We will talk about faster approaches to reach
  global points in future under the topics such as
  PSO

Global optimal point
29
Constrained optimization methods

• For example consider the following problem
• If this were unconstrained, the optimal point
  would be
• For the constrained problem this point is
  unfeasible
• how do we solve the problem?.