Goal Programming

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Transcript of Goal Programming

Chapter 15Goal 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 DMs 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

Goal Programming Constraints

Economic Constraints

Stated as =, or = Linear (stated in terms of decision variables) Example: 3x + 2y 0 Current Clients: X1 + d1- - d1+ = 200 New Clients: X2 + d2- - d2+ = 120

Goal Constraints: 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- + d2Subject to: 2X1 + 3X2 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 SolutionX2(0,213)

200 150 100 50(320,0)

0

50

100

150

200 250 300 350

X1

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 ExampleX2(0,213)

Goal 1 d1d1+ d2+ Goal 2(140,120) (200,80)

200 150 100 50

d2-

(320,0)

0

50

100

150

X1 200 250 300 350

Solving Graphical Goal Programming

Want to Minimize d1- + d2So we evaluate each of the candidate solution points:For point (140, 120) d1- = 60 and d2- = 0 Z = 60 + 0 = 60

Optimal Point

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

Goal Programming Solution

X1 = 200 X2 = 80 d1+ = 0 d1- = 0 Z = 40

Goal 1 achieved Goal 2 not achieved d2+ = 0 d2- = 40

For Next Class

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