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Chapter 15
Goal Programming
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 DM’s utility for meeting the various goals.
Assumptions
Similar to LP: Non-negative variables Conditions of certainty Variables are independent Limited resources Deterministic
Components
Economic Constraints Physical Concerned with resources Cannot be violated Example: # of production hours each week
Components
Goal Constraints Variable Concerned with target values Can be changed/modified Example: Desire to achieve a certain level of
profit
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)
Goal Programming Steps
Define decision variables Define Deviational Variable for each goal Formulate Constraint Equations
Economic constraints Goal constraints
Formulate Objective Function
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
Economic Constraints Stated as <=, >=, or = Linear (stated in terms of decision variables) Example: 3x + 2y <= 50 hours
Goal Constraints General form of goal constraint:
- d+ + d- =
Goal Programming Constraints
Decision Variables
Desired Goal Level
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
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
Goal Programming Example
Steps Required:1. Define the decision variables
2. Define the goals and deviational variables
3. Formulate the GP Model’s Parameters: Economic Constraints Goal Constraints Objective Function
4. Solve the GP using the graphical approach
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
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
Goal Programming Example
Formulate the GP Model: Economic Constraints:
2X1 + 3X2 <= 640 (note: can be <, =, >) X1, X2 => 0 d1+, d1-, d2+, d2- => 0
Goal Constraints: Current Clients: X1 + d1- - d1+ = 200 New Clients: X2 + d2- - d2+ = 120
Must be =
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
Goal Programming Example
Objective Function: Minimize Weighted Deviations Minimize Z = d1- + d2-
Goal Programming Example
Complete formulation: Minimize Z = d1- + d2-
Subject to: 2X1 + 3X2 <= 640 X1 + d1- - d1+ = 200 X2 + d2- - d2+ = 120 X1, X2 => 0 d1+, d1-, d2+, d2- => 0
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)
Graphical Solution
00 5050 100100 150150 200200
5050
100100
150150
200200
XX22
2X2X11 + 3X + 3X
22 = 640
= 640
250250 300300 350350
(0,213)(0,213)
(320,0)(320,0)XX11
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
Goal Programming Example
00 5050 100100 150150 200200
5050
100100
150150
200200
XX11
XX22 Goal 1Goal 1
dd11--
dd11++
Goal 2Goal 2Goal 2Goal 2dd22
++dd22++
dd22--dd22--(140,120)(140,120)
(200,80)(200,80)
2X2X11 + 3X + 3X
22 < = 640
< = 640
250250 300300 350350
(0,213)(0,213)
(320,0)(320,0)
Solving Graphical Goal Programming
Want to Minimize d1- + d2-
So we evaluate each of the candidate solution points:
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
Optimal Point
Contact at least 200 current clientsContact at least 120 new clients
Goal Programming Solution
X1 = 200 Goal 1 achieved X2 = 80 Goal 2 not achieved d1+ = 0 d2+ = 0 d1- = 0 d2- = 40
Z = 40
For Next Class
Complete reading Goal Programming pages (thru 727&
Do Goal Programming HWs