Goal Programming
Multiple-Objectives Problem: In most practical cases, decision
makers are faced a situation where they must achieve more
than two objectives (those may even be in conflict) at same
time.
Or more than two criteria must be used to evaluate a decision.
Examples:
Production Planning - Maximize Profit/Maximize Market
Share
Location Selection - Maximize Sales/Minimize Delivery Cost
Personal Schedule - Maximize GPA/Maximize Income
Formulation of GP Problems
Deviations: the amount away from the desired standards or objectives:
– Overachievement (d+i ≥ 0) vs. Underachievement (d-i ≥ 0)
– Desirable vs. Undesirable Deviations: (depend on the objectives)
• Max goals (≥) - the more the better - d+i desirable.
• Min goals (≤) - the less the better - d-i desirable.
• Exact goals (=) - exactly equal - both d+i and d-i undesirable
– In GP, the objective is to minimize the (weighted) sum of
undesirable deviations (all undesirable d+i and d-i →→ 0 ).
– For each goal, at least, one of d+i and d-i must be equal to "0"
Formulation of GP Problems
• Goals are prioritized in some sense, and their level of aspiration
is stated.
• An optimal solution is attained when all the goals are reached as
close as possible to their aspiration level, while satisfying a set
of constraints.
• There are two types of goal programming models:
– Nonpreemptive goal programming - no goal is pre-determined to
dominate any other goal.
– Preemptive goal programming - goals are assigned different priority
levels. Level 1 goal dominates level 2 goal, and so on.
NONPREEMPTIVE GOAL PROGRAMMING
An Advertisement Example
• A company is considering three forms of advertising.
Cost per Ad Customers
• Goals
Television
Radio
Newspaper
3000
800
250
1000
500
200
– Goal 1: Spend no more $25,000 on advertising.
– Goal 2: Reach at least 30,000 new potential customers.
– Goal 3: Run at least 10 television spots.
An Advertisement Example
LP Model:
3000X1 + 800X2 + 250X3  25,000
1000X1 + 500X2 + 200X3  30,000
X1
 10
An Advertisement Example
• Detrimental variables
Ui = the amount by which the left hand side falls short
of (under) its right hand side value.
Ei = the amount by which the left hand side exceeds its
right hand side value.
• The goal equations
3000X1 + 800X2 + 250X3 + U1 – E1 = 25,000
1000X1 + 500X2 + 200X3 + U2 – E2 = 30,000
X1
+ U3 – E3 = 10
An Advertisement Example
• The objective is to minimize the penalty of not meeting the
goals, represented by the detrimental variables
E1 U2 U3
 25,000
 30,000
 10
An Advertisement Example
• The penalties are estimated to be as follows:
– Each extra dollar spent on advertisement above $25,000
cost the company $1.
– There is a loss of $5 to the company for each customer not
being reached, below the goal of 30,000.
– Each television spot below 10 is worth 100 times each
dollar over budget.
An Advertisement Example –
The goal programming model
• It is assumed that no advantage is gained by overachieving a
goal.
Minimize 1E1 + 5U2 + 100U3
s.t.
3000X1 + 800X2 + 250X3 + U1 – E1 = 25,000
1000X1 + 500X2 + 200X3 + U2 – E2 = 30,000
X1
+ U3 – E3 = 10
All variables are non-negative.
NONPREEMPTIVE GOAL PROGRAMMING
Conceptual Products
Conceptual Products is a computer company that produces the
CP400 and the CP500 computers. The computers use different
mother boards produced in abundant supply by the company, but use
the same cases and disk drives. The CP400 models use two floppy
disk drives and no zip disk drives whereas the CP500 models use one
floppy disk drive and one zip disk drive. The disk drives and cases
are bought from vendors. There are 1000 floppy disk drives, 500 zip
disk drives, and 600 cases available to Conceptual Products on a
weekly basis. It takes one hour to manufacture a CP400 and its profit
is $200 and it takes one and one-half hours to manufacture a CP500
and its profit is $500.
Goals:
• Goal 1: Produce at least 200 CP400 computers
each week.
• Goal 2: Produce at least 500 total computers
each week.
• Goal 3: Reach at least $250 (in thousands) on
profit.
• Goal 4: Consume no more than 400 total manhours each week.
NONPREEMPTIVE GOAL PROGRAMMING
Conceptual Products
NONPREEMPTIVE GOAL PROGRAMMING
Conceptual Products
NONPREEMPTIVE GOAL PROGRAMMING
Conceptual Products
NONPREEMPTIVE GOAL PROGRAMMING
Conceptual Products
PREEMPTIVE GOAL PROGRAMMING
PREEMPTIVE GOAL PROGRAMMING
PREEMPTIVE GOAL PROGRAMMING
PREEMPTIVE GOAL PROGRAMMING
PREEMPTIVE GOAL PROGRAMMING
PREEMPTIVE GOAL PROGRAMMING
PREEMPTIVE GOAL PROGRAMMING
PREEMPTIVE GOAL PROGRAMMING
PREEMPTIVE GOAL PROGRAMMING