Download - Goal Programming

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


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


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


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


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


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 =


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


Objective Function: Minimize Weighted Deviations Minimize Z = d1- + d2-


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


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


Graph deviation lines X1 + d1- - d1+ = 200 (Goal 1) X2 + d2- - d2+ = 120 (Goal 2)

Plot lines for X1 = 200, X2 = 120


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

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