Goal Programming

1
Chapter 15

• Goal Programming

2
What is Goal Programming?

• Mathematical model similar to Linear Programming,
  however it allows for multiple goals to be
  satisfied at the same time.
• Allows for the multiple goals to be prioritized
  and weighted to account for the DMs utility for
  meeting the various goals.

3
Assumptions

• Similar to LP
• Non-negative variables
• Conditions of certainty
• Variables are independent
• Limited resources
• Deterministic

4
Components

• Economic Constraints
• Physical
• Concerned with resources
• Cannot be violated
• Example of production hours each week

5
Components

• Goal Constraints
• Variable
• Concerned with target values
• Can be changed/modified
• Example Desire to achieve a certain level of
  profit

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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 was maximize
  revenue/minimize costs)

7
Goal Programming Steps

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

8
Goal Programming Terms

• Decision Variables are the same as those in LP
  formulations (represent products, hours worked)
• 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

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Goal Programming Constraints

• Economic Constraints
• Stated as lt, gt, or
• Linear (stated in terms of decision variables)
• Example 3x 2y lt 50 hours
• Goal Constraints
• General form of goal constraint
• - d d-

Decision Variables
Desired Goal Level
10
Goal Programming Example

• Microcom is a growth oriented firm which
  establishes monthly performance goals for its
  sales force
• Microcom determines that the sales force has a
  maximum available hours per month for visits of
  640 hours
• Further, it is estimated that each visit to a
  potential new client requires 3 hours and each
  visit to a current client requires 2 hours

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Goal Programming Example

• Microcom establishes two goals for the coming
  month
• Contact at least 200 current clients
• Contact at least 120 new clients
• Overachieving either goal will not be penalized

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Goal Programming Example

• Steps Required
• Define the decision variables
• Define the goals and deviational variables
• Formulate the GP Models Parameters
• Economic Constraints
• Goal Constraints
• Objective Function
• Solve the GP using the graphical approach

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Goal Programming Example

• Step 1 Define the decision variables
• X1 the number of current clients visited
• X2 the number of new clients visited
• Step 2 Define the goals
• Goal 1 Contact 200 current clients
• Goal 2 Contact 120 new clients

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Goal Programming Example

• Step 3 Define the deviational variables
• d1 the number of current clients visited in
  excess of the goal of 200
• d1- the number of current clients visited less
  than the goal of 200
• d2 the number of new clients visited in excess
  of the goal of 120
• d2- the number of new clients visited less than
  the goal of 120

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Goal Programming Example

• Formulate the GP Model
• Economic Constraints
• 2X1 3X2 lt 640 (note can be lt, , gt)
• X1, X2 gt 0
• d1, d1-, d2, d2- gt 0
• Goal Constraints
• Current Clients X1 d1- - d1 200
• New Clients X2 d2- - d2 120

Must be
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Goal Programming Example

• WebNet establishes two goals for the coming
  month
• Contact at least 200 current clients
• Contact at least 120 new clients
• Overachieving either goal will not be penalized

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Goal Programming Example

• Objective Function
• Minimize Weighted Deviations
• Minimize Z d1- d2-

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Goal Programming Example

• Complete formulation
• Minimize Z d1- d2-
• Subject to
• 2X1 3X2 lt 640
• X1 d1- - d1 200
• X2 d2- - d2 120
• X1, X2 gt 0
• d1, d1-, d2, d2- gt 0

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Goal Programming Example

• Graph constraint
• 2X1 3X2 640
• If X1 0, X2 213
• If X2 0, X1 320
• Plot points (0, 213) and (320, 0)

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Graphical Solution
X2
(0,213)
200
150
2X1 3X2 640
100
50
(320,0)
X1
0
50
100
150
200
250
300
350
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Goal Programming Example

• Graph deviation lines
• X1 d1- - d1 200 (Goal 1)
• X2 d2- - d2 120 (Goal 2)
• Plot lines for X1 200, X2 120

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Goal Programming Example
X2
Goal 1
(0,213)
200
d1-
2X1 3X2 lt 640
d1
150
d2
Goal 2
(140,120)
100
d2-
(200,80)
50
(320,0)
X1
0
50
100
150
200
250
300
350
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Solving Graphical Goal Programming

• Want to Minimize d1- d2-
• So we evaluate each of the candidate solution
  points

Optimal Point
For point (140, 120) d1- 60 and d2- 0 Z
60 0 60
For point (200, 80) d1- 0 and d2- 40 Z
0 40 40
Contact at least 200 current clients Contact at
least 120 new clients
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Goal Programming Solution

• X1 200 Goal 1 achieved
• X2 80 Goal 2 not achieved
• d1 0 d2 0
• d1- 0 d2- 40
• Z 40

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For Next Class

• Complete reading Goal Programming pages (thru
  727
• Do Goal Programming HWs