Goal Programming In many linear programming problems, the objective function or goal extends beyond...

Goal ProgrammingGoal Programming

In many linear programming problems, theobjective function or goal extends beyond just maximizing total profit or minimizing

total cost.

Applied Management Science for Decision Making, 2e Applied Management Science for Decision Making, 2e © 2014 Pearson Learning Solutions Philip A. Vaccaro , PhD© 2014 Pearson Learning Solutions Philip A. Vaccaro , PhD

Strategic Resource Allocation

and PlanningMGMT E-5050


Maximizing market shareMaximizing market share

Maintaining full employmentMaintaining full employment

Minimizing production idle Minimizing production idle timetime

Restricting overtime laborRestricting overtime labor

Adhering to limited storage Adhering to limited storage spacespace

POSSIBLE ECONOMIC GOALSPOSSIBLE ECONOMIC GOALS

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


Maximizing land usageMaximizing land usage

Minimizing federal grant Minimizing federal grant overspendingoverspending

Maximizing the number of Maximizing the number of people reached by adverti-people reached by adverti-singsing

Minimizing noise levels in Minimizing noise levels in the neighborhoodthe neighborhood

Minimizing staff shortagesMinimizing staff shortages

POSSIBLE NON-ECONOMIC GOALSPOSSIBLE NON-ECONOMIC GOALS


Regardless of their nature and number, these multiple andRegardless of their nature and number, these multiple and diverse goals share several common characteristics:diverse goals share several common characteristics:

THEY CANNOT BE MEASURED ON THE

SAME SCALE

SOME INVOLVE SQUAREFOOTAGE, MONEY, TIME,

COMPLIANCE, AND QUALITY OF LIFE ISSUES

THEY ARE USUALLYIN CONFLICT

THE ACHIEVEMENT OFONE OBJECTIVE

THREATENS DIMINISHMENTOR ABANDONMENT

OF ANOTHER

Multiple ObjectivesMultiple ObjectivesTWO APPROACHES

1st Approach1st Approach

DEVELOP A SINGLE DEVELOP A SINGLE OBJECTIVEOBJECTIVEFUNCTIONFUNCTION THAT ADDRESSES THAT ADDRESSES

ALL GOALS SIMULTANEOUSLYALL GOALS SIMULTANEOUSLY

THIS IS ACCOMPLISHEDTHIS IS ACCOMPLISHEDBY CONVERTING THEBY CONVERTING THE

VALUES OF ALL THE GOALSVALUES OF ALL THE GOALSTO A COMMON MEASURE OFTO A COMMON MEASURE OF

VALUE OR UTILITYVALUE OR UTILITY

Multiple ObjectivesMultiple ObjectivesTWO APPROACHES

2nd Approach2nd Approach

ACKNOWLEDGE THEACKNOWLEDGE THEDIFFERENCES IN THEDIFFERENCES IN THEVARIOUS GOALS ANDVARIOUS GOALS ANDUSE A PROCEDURE,USE A PROCEDURE,EITHER EITHER GRAPHICALGRAPHICAL,,COMPUTER-BASED,COMPUTER-BASED,

OR OR SIMPLEX, SIMPLEX, TOTO ADDRESS THEMADDRESS THEM

INDIVIDUALLYINDIVIDUALLY

THIS IS CALLEDTHIS IS CALLEDGOAL PROGRAMMINGGOAL PROGRAMMING

Goal ProgrammingGoal ProgrammingTHE PRIMARY PURPOSETHE 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 actualdeviations from those desired levels.

Goal vs. Linear ProgrammingGoal vs. Linear Programming

THE DIFFERENCESTHE 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 ONLYONLY variables in the objective function.

The objective function is formulated to MINIMIZEMINIMIZE the sum of the differences between the deviational variables.


THERE MAY BE GOALS THATMUST BE METMUST BE MET , AS OPPOSED

TO GOALS THAT THE DECISION MAKER ATTEMPTS

TO MEET

THE FORMER WILL ALWAYS BEREPRESENTED BY REGULAR

LINEAR PROGRAMMINGCONSTRAINTS !

Goal ProgrammingGoal ProgrammingTHE OBJECTIVE FUNCTIONTHE OBJECTIVE FUNCTION

A function of the deviational variables, it representsnon-achievement of the various established goals,

and must always be minimized

deviational variable that measures under achievement

deviational variable that measures over achievement

d

d

-

+

Goal ProgrammingGoal ProgrammingINDICATORS

Words such as :Words such as :

…….or others that imply that there.or others that imply that there is a level of performance , belowis a level of performance , below which, or above which, one doeswhich, or above which, one does not want to be.not want to be.

• shouldshould• mustmust• oughtought• avoidavoid• exactlyexactly

Goal ProgrammingGoal ProgrammingOBJECTIVE FUNCTION FORMULATIONOBJECTIVE 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 to be minimized.

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

Goal Programming ExampleGoal Programming ExampleHARRISON ELECTRIC COMPANYHARRISON ELECTRIC COMPANY

The Harrison Electric Company produces two products popularwith 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 towire a ceiling fan. Final assembly of the chandeliers and fansrequires 6 and 5 hours, respectively. The production capabilityis such that only 12 hours of wiring time and 30 hours of assem-bly time are available. Each chandelier nets the firm $7.00 andeach fan $6.00 .

REQUIREMENT:

1. Formulate the objective function and resource constraints.

PROBLEM STATEMENT


Maximize Z = $7.00 X1 + $6.00 X2

subject to:

2X1 + 3X2 =< 12 wiring hours

6X1 + 5X2 =< 30 assembly hours

Where: XWhere: X11 = CHANDELIERS PRODUCED = CHANDELIERS PRODUCED XX22 = CEILING FANS PRODUCED = CEILING FANS PRODUCED

MODEL FORMULATION

OBJECTIVE OBJECTIVE FUNCTIONFUNCTION

RESOURCERESOURCECONSTRAINTSCONSTRAINTS


OBJECTION FUNCTION MODIFICATION

Management now wants to achieve a profit goal Management now wants to achieve a profit goal ofof exactlyexactly $30.00. The new objective function is: $30.00. 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


SINGLE GOAL MODEL FORMULATION

Minimize total deviation = d1 + d1+-

subject to:

7X1 + 6X2 + d1 - d1 = $30.00 profit goal

2X1 + 3X2 =< 12 wiring hours

6X1 + 5X2 =< 30 assembly hours

X1, X2, d1- , d1+ => 0 NON-NEGATIVITY CONSTRAINT

- +


SINGLE GOAL MODEL SOLUTIONSINGLE GOAL MODEL SOLUTION

X1 ( chandeliers ) = 4.2857 units

X2 ( ceiling fans ) = 0 units

d1 = 0 ( no profit overachievement )

d1 = 0 ( no profit underachievement )

d2 = 3.4286 ( under-used wiring hours )

d3 = 4.2857 ( under-used assembly hrs )

Z = 0 ( objective function )

DETERMINEDVIA

COMPUTER++

--

--

--


X1 ( chandeliers ) = 4.2857 units

X2 ( ceiling fans ) = 0 units

d1 = 0 ( no profit overachievement )

d1 = 0 ( no profit underachievement )

d2 = 3.4286 ( under-used wiring hours )

d3 = 4.2857 ( under-used assembly hrs )

Z = 0 ( objective function )

IF THE TARGETGOAL OF$30.00 IS

ACHIEVEDEXACTLY, BOTH

d1+ AND d1-WILL BE

EQUAL TO ZERO

THE OBJECTIVE FUNCTION WILL ALSO

BE MINIMIZED

AT ZERO

+

-

-

-

SINGLE GOAL MODEL SOLUTION


If overachievement were acceptable, the “d+” deviational variable can be eliminated from the objective function.

If underachievement were acceptable, the “d-” deviational variable can be eliminated from the objective function.

SOLUTION POSTSCRIPT


GOAL RANKING SCENARIO

Produce as much profit above $30.00 asProduce as much profit above $30.00 aspossible during the production period.possible during the production period.

Fully employ available wiring hours.Fully employ available wiring hours.

Avoid overtime in assembly hours.Avoid overtime in assembly hours.

Produce Produce at leastat least seven (7) ceiling fans. seven (7) ceiling fans.

Goal #2

Goal #3

Goal #4Goal #4

Goal #1


GOAL RANKING THE VARIABLES

PP11dd1+/-1+/-

PP22dd2+/-2+/-

PP33dd3+/- 3+/-

PP44dd4+/- 4+/-

THE RANKINGSTHE RANKINGS

profit target

wiring hours used

assembly hours used

ceiling fans produced


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

-

-

-

-

+

+

+

+


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.


WEIGHTED GOALS

If goal # 4 is the If goal # 4 is the least important goalleast important goal……………… its weight is “……………… its weight is “1””

If goal # 3 is If goal # 3 is twice as importanttwice as important as goal # 4…….. ..its weight is “ as goal # 4…….. ..its weight is “2””

If goal # 2 is If goal # 2 is four times as importantfour times as important as goal # 4... its weight is “ as goal # 4... its weight is “4””

If goal # 1 is If goal # 1 is six times as importantsix times as important as goal # 4… ..its weight is “ as goal # 4… ..its weight is “6” ”

THE LEAST IMPORTANT GOAL IS ALWAYS WEIGHTED “1”


WEIGHTED GOALS

THE OBJECTIVE FUNCTION BECOMES….THE OBJECTIVE FUNCTION BECOMES….

Minimize Minimize ΣΣ deviations = deviations = 66dd11 + + 44dd22 + + 22dd3 3 + + 11dd44-- ---- ++

THE RELATIVE WEIGHTSTHE RELATIVE WEIGHTS


GOAL RANKING WITH WEIGHTS

THE OBJECTIVE FUNCTION BECOMES….THE OBJECTIVE FUNCTION BECOMES….

Minimize Minimize ΣΣ deviations = deviations = PP11( ( 66dd1 1 ) + ) + PP22( ( 44dd2 2 ) + ) + PP33( ( 22dd3 3 ) + ) + PP44( ( 11dd4 4 ))-- ---- ++

THE RANKINGSTHE RANKINGS

THE WEIGHTSTHE WEIGHTS

Goal Programming with QM for WINDOWS

Harrison ElectricCompany

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.

Goal Programming ExampleAttachAttachéé Training Program Training Program

Major Bill Bligh, Director of the Army War College’s newsix-month attaché training program, is concerned abouthow the 20 officers taking the course spend their timewhile in his charge. Major Bligh recognizes that there are 168 hours per weekand thinks that his students have been using them ratherinefficiently. 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 )

Goal Programming ExampleGoal Programming ExampleAttachAttachéé Training Program Training Program

He thinks that students should study 30 hours a weekto have time to absorb the material. This is his mostimportant goal. Bligh feels that students need at most 7 hours sleep per night on average and that this goalis number 2.He believes that goal number 3 is to provide at least 20 hours per week of social time.

Requirement:

1. Formulate this as a goal programming problem.

2. Solve the problem using computer software.


MODEL FORMULATION

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

Let:


MODEL FORMULATION

The objective function becomes:

Minimize = d1- + d1+ + d2+ + 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

personaltime is

whateveris left !

( “d4” can beomitted )


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 = P1d1- + P1d1+ + P2d2+ + P3d3-

where:

P1 = meet class and study goal

P2 = meet sleeping goal

P3 = meet socializing goal

All goals are fully met !

Goal Programming ExampleGoal Programming ExampleAttachAttachéé Training Program Training Program

MODEL SOLUTIONMODEL SOLUTION

X1 = 49 hours, sleep

X2 = 69 hours, personal

X3 = 30 hours, class and studying

X4 = 20 hours, social time

Goal Programming with QM for WINDOWS

Attache TrainingProgram

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

Goal Programming In many linear programming problems, the objective function or goal extends beyond...

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