1. Goal Programming In many linear programming problems, the
Strategic Resource
Allocation
and Planning
MGMT E-5050
In many linear programming problems, the
objective function or goal extends beyond
just maximizing total profit or minimizing
total cost.
Applied Management Science for Decision Making, 2e © 2014 Pearson Learning Solutions Philip A. Vaccaro , PhD

2. Goal Programming Maximizing market share Maintaining full employment
POSSIBLE ECONOMIC GOALS
Maximizing market share
Maintaining full employment
Minimizing production idle time
Restricting overtime labor
Adhering to limited storage space
Applied Management Science for Decision Making, 2e © 2014 Pearson Learning Solutions

3. Goal Programming Maximizing land usage
POSSIBLE NON-ECONOMIC GOALS
Maximizing land usage
Minimizing federal grant overspending
Maximizing the number of people reached by adverti-sing
Minimizing noise levels in the neighborhood
Minimizing staff shortages

4. Goal Programming
Regardless of their nature and number, these multiple and
diverse goals share several common characteristics:
THEY CANNOT BE
MEASURED ON THE SAME SCALE
SOME INVOLVE SQUARE FOOTAGE, MONEY, TIME,
COMPLIANCE, AND
QUALITY OF LIFE ISSUES
THEY ARE USUALLY
IN CONFLICT
THE ACHIEVEMENT OF ONE OBJECTIVE
THREATENS DIMINISHMENT
OR ABANDONMENT
OF ANOTHER

5. Multiple Objectives 1st Approach TWO APPROACHES
DEVELOP A SINGLE OBJECTIVE
FUNCTION THAT ADDRESSES
ALL GOALS SIMULTANEOUSLY
THIS IS ACCOMPLISHED
BY CONVERTING THE
VALUES OF ALL THE GOALS
TO A COMMON MEASURE OF
VALUE OR UTILITY

6. Multiple Objectives 2nd Approach TWO APPROACHES ACKNOWLEDGE THE
DIFFERENCES IN THE
VARIOUS GOALS AND
USE A PROCEDURE,
EITHER GRAPHICAL,
COMPUTER-BASED,
OR SIMPLEX, TO
ADDRESS THEM
INDIVIDUALLY
THIS IS CALLED
GOAL PROGRAMMING

7. Goal Programming Not to develop an optimal solution as such,
THE PRIMARY PURPOSE
Not to develop an optimal solution as such,
but to ‘satisfice’, that is, meet the desired level
of achievement set for each goal, and if that is
not entirely possible, to minimize the actual
deviations from those desired levels.

8. Goal vs. Linear Programming
THE DIFFERENCES
Goal programming attempts to minimize the deviations
between the various established goals and what can be
actually achieved, given the available resources.
Variables called deviational variables are typically the
ONLY variables in the objective function.
The objective function is formulated to MINIMIZE the sum
of the differences between the deviational variables.

9. Goal Programming THERE MAY BE GOALS THAT MUST BE MET , AS OPPOSED
TO GOALS THAT THE
DECISION MAKER ATTEMPTS
TO MEET
THE FORMER WILL ALWAYS BE
REPRESENTED BY REGULAR LINEAR PROGRAMMING CONSTRAINTS !

10. Goal Programming
THE OBJECTIVE FUNCTION
A function of the deviational variables, it represents
non-achievement of the various established goals,
and must always be minimized - d
deviational variable that measures under achievement + d
deviational variable that measures over achievement

11. Goal Programming Words such as : INDICATORS should must ought avoid
exactly
….or others that imply that there
is a level of performance , below
which, or above which, one does
not want to be.

12. Goal Programming * * If overachievement is undesired, the deviational
OBJECTIVE FUNCTION FORMULATION
If overachievement is undesired, the deviational
variable d+ will be placed in the objective function
to be minimized.
If underachievement is undesired, the deviational
variable d- will be placed in the objective function
If both overachievement and underachievement are
undesired, the deviational variables d+ and d- will be
placed in the objective function to be minimized. * *
THAT IS, THE GOAL IS TO BE MET EXACTLY

13. Goal Programming Example
HARRISON ELECTRIC COMPANY
The Harrison Electric Company produces two products popular
with home renovators: old-fashioned chandeliers and ceiling
fans Both products require a two-step production process
involving wiring and assembly.
It takes about 2 hours to wire each chandelier and 3 hours to
wire a ceiling fan. Final assembly of the chandeliers and fans
requires 6 and 5 hours, respectively. The production capability
is such that only 12 hours of wiring time and 30 hours of assem-
bly time are available. Each chandelier nets the firm $7.00 and
each fan $6.00 .
REQUIREMENT:
Formulate the objective function and resource constraints.
PROBLEM STATEMENT

14. Goal Programming Example
HARRISON ELECTRIC COMPANY
OBJECTIVE
FUNCTION
Maximize Z = $7.00 X1 + $6.00 X2
subject to:
2X1 + 3X2 =< wiring hours
6X1 + 5X2 =< 30 assembly hours
MODEL FORMULATION
RESOURCE
CONSTRAINTS
Where: X1 = CHANDELIERS PRODUCED
X2 = CEILING FANS PRODUCED

15. Goal Programming Example
HARRISON ELECTRIC COMPANY
OBJECTION FUNCTION MODIFICATION
Management now wants to achieve a profit goal
of exactly $ The new objective function is: - +
Minimize deviations = d1 + d1
WHERE d1- REPRESENTS UNDERACHIEVEMENT OF THE PROFIT GOAL
WHERE d1+ REPRESENTS OVERACHIEVEMENT OF THE PROFIT GOAL

16. Goal Programming Example
HARRISON ELECTRIC COMPANY - +
Minimize total deviation = d1 + d1
subject to:
SINGLE GOAL MODEL FORMULATION - +
7X1 + 6X2 + d1 - d1 = $ profit goal
2X1 + 3X2 =< wiring hours
6X1 + 5X2 =< assembly hours
X1, X2, d1- , d1+ => NON-NEGATIVITY CONSTRAINT

17. Goal Programming Example
HARRISON ELECTRIC COMPANY
SINGLE GOAL MODEL SOLUTION
X1 ( chandeliers ) = units
X2 ( ceiling fans ) = 0 units
d1 = 0 ( no profit overachievement )
d1 = 0 ( no profit underachievement )
d2 = ( under-used wiring hours )
d3 = ( under-used assembly hrs )
Z = 0 ( objective function )
DETERMINED
VIA
COMPUTER + - - -

18. Goal Programming Example
HARRISON ELECTRIC COMPANY
X1 ( chandeliers ) = units
X2 ( ceiling fans ) = 0 units
d1 = 0 ( no profit overachievement )
d1 = 0 ( no profit underachievement )
d2 = ( under-used wiring hours )
d3 = ( under-used assembly hrs )
Z = 0 ( objective function )
IF THE TARGET
GOAL OF
$30.00 IS
ACHIEVED
EXACTLY, BOTH
d1+ AND d1-
WILL BE
EQUAL TO
ZERO
SINGLE GOAL MODEL SOLUTION + -
THE OBJECTIVE
FUNCTION
WILL ALSO BE
MINIMIZED
AT ZERO - -

19. Goal Programming Example
HARRISON ELECTRIC COMPANY
SOLUTION POSTSCRIPT
If overachievement were acceptable, the “d+”
deviational variable can be eliminated from
the objective function.
If underachievement were acceptable, the “d-”

20. Goal Programming Example
HARRISON ELECTRIC COMPANY
GOAL RANKING SCENARIO
Goal #1
Produce as much profit above $30.00 as
possible during the production period.
Fully employ available wiring hours.
Avoid overtime in assembly hours.
Produce at least seven (7) ceiling fans.
Goal #2
Goal #3
Goal #4

21. Goal Programming Example
HARRISON ELECTRIC COMPANY
GOAL RANKING THE VARIABLES
P1d1+/-
P2d2+/-
P3d3+/-
P4d4+/-
profit target
wiring hours used
assembly hours used
ceiling fans produced
THE RANKINGS

22. Goal Programming Example
HARRISON ELECTRIC COMPANY
THE GOAL RANKING MODEL - - + -
Minimize Σ deviations = P1d1 + P2d2 + P3d3 + P4d4
subject to: + -
7X1 + 6X2 + d1 – d1 = 30 ( profit )
2X1 + 3X2 + d2 – d2 = 12 ( wiring hours )
6X1 + 5X2 + d3 – d3 = 30 ( assembly hours )
1X2 + d4 – d4 = 7 ( ceiling fans ) - + - + - +
All Xi , di variables => 0

23. Goal Programming Example
HARRISON ELECTRIC COMPANY
WEIGHTED GOALS
Sometimes, one goal is more important than
another goal, but their priority levels are the
very same.
In these cases, the coefficients in the objective
function for the deviational variables are the
weights assigned to the goals.

24. Goal Programming Example
HARRISON ELECTRIC COMPANY
WEIGHTED GOALS
If goal # 4 is the least important goal……………… its weight is “1”
If goal # 3 is twice as important as goal # 4…….. ..its weight is “2”
If goal # 2 is four times as important as goal # 4... its weight is “4”
If goal # 1 is six times as important as goal # 4… ..its weight is “6”
THE LEAST IMPORTANT GOAL IS ALWAYS WEIGHTED “1”

25. Goal Programming Example
HARRISON ELECTRIC COMPANY
WEIGHTED GOALS
THE OBJECTIVE FUNCTION BECOMES…. - - + -
Minimize Σ deviations = 6d1 + 4d2 + 2d3 + 1d4
THE RELATIVE WEIGHTS

26. Goal Programming Example
HARRISON ELECTRIC COMPANY
GOAL RANKING WITH WEIGHTS
THE OBJECTIVE FUNCTION BECOMES….
THE RANKINGS - - + -
Minimize Σ deviations = P1( 6d1 ) + P2( 4d2 ) + P3( 2d3 ) + P4( 1d4 )
THE WEIGHTS

27. Goal Programming with QM for WINDOWS
Harrison Electric
Company

28. Applied Management Science for Decision Making, 2e © 2014 Pearson Learning Solutions

31. Goal 1 : to produce profit of $30
Goal 1 : to produce profit of $30.00 if possible during the production period.
Goal 2 : to fully utilize the available wiring department hours.
Goal 3 : to avoid overtime in the assembly department.
Goal 4 : to meet a contract requirement to produce at least seven ceiling fans.

36. Goal Programming Example
Attaché Training Program
Major Bill Bligh, Director of the Army War College’s new
six-month attaché training program, is concerned about
how the 20 officers taking the course spend their time
while in his charge.
Major Bligh recognizes that there are 168 hours per week
and thinks that his students have been using them rather
inefficiently. Bligh lets:
X1 = number of hours of sleep needed per week
X2 = number of personal hours ( eating, personal
hygiene, handling laundry, and so on ).
X3 = number of hours of class and studying.
X4 = number of hours of social time off base ( dating,
sports, family visits, and so on )

37. Goal Programming Example
Attaché Training Program
He thinks that students should study 30 hours a week
to have time to absorb the material. This is his most
important goal. Bligh feels that students need at most
7 hours sleep per night on average and that this goal
is number 2.
He believes that goal number 3 is to provide at least
20 hours per week of social time.
Requirement:
Formulate this as a goal programming problem.
Solve the problem using computer software.

38. Goal Programming Example
Attaché Training Program
MODEL FORMULATION
Let:
d1- = underachievement of class and study goal
d1+ = overachievement of class and study goal
d2+ = overachievement of sleeping goal
d3- = underachievement of social time goal

39. Goal Programming Example
Attaché Training Program
MODEL FORMULATION
The objective function becomes:
Minimize = d d d d3-
subject to constraints ( per week )
1X3 + d1- - d1+ = 30 hours class / study
1X1 + d2- - d2+ = 49 hours sleep
1X4 + d3- - d3+ = 20 hours social time
all variables => 0
1X1 + 1X2 + 1X3 + 1X4 =< 168 hours
personal
time is
whatever
is left !
( “d4” can be
omitted )

40. Goal Programming Example
Attaché Training Program
MODEL FORMULATION
Since the goals have priority, they can be rewritten in this
order, yielding the absolute completion of each goal before
attempting to achieve the next goal.
The objective function would become:
Minimize = P1d P1d P2d P3d3-
where:
P1 = meet class and study goal
P2 = meet sleeping goal
P3 = meet socializing goal

41. Goal Programming Example
Attaché Training Program
MODEL SOLUTION
X1 = 49 hours, sleep
X2 = 69 hours, personal
X3 = 30 hours, class and studying
X4 = 20 hours, social time
All goals are fully met !

42. Goal Programming with QM for WINDOWS
Attache Training
Program

45. Goal 1 , 1st Priority : Class and Study time should be 30 hours
Goal 2 , 2nd Priority : Sleep time must be, at most, 49 hours
Goal 3 , 3rd Priority : Social time must be at least 20 hours

50. Goal Programming