Goal Programming

GETTING DRESSED 5 Jackets, 10 Pants, 20 Ties, 10 Shirts, 10 Pairs of Socks, 4 Pairs of

Shoes, 5 Belts

2,000,000 Possible Outfits!

Takes 1 Second To Evaluate Each Outfit

How Long To Get Dressed?

23.1 Days!

+ Colour matching

Personal favourites/likes and dislikes

Theme of the occasion

My status in society

Other’s opinion/interest

……….etc (end of thinking capacity)

OPTIMAL SOLUTIONS

• Car with low cost

• Good mileage

• Esthetics

• Power

• Comfort

• Safety

Buying a new car

Selecting a college for your MBA

• College with good reputation

• Low tuition fee

• Close to home

• Right program

They may want to achieve several, sometimes contradictory, goals

Location Selection - Maximize Sales/Minimize Delivery Cost

Maximize profit - avoid employee layoffs - danger of a labour strike

Faster delivery but also - higher quality

Expand while becoming more efficient

Satisficing

In reality, it may be impossible to satisfy all objectives at

their highest degree at the same time.

Instead, tradeoff and compromise must be made.

That is, there is only "satisfied" solutions, no "optimal"

solutions.

Satisficing

It is a discipline aimed at supporting decision

makers faced with numerous and sometimes

conflicting evaluations

• GOAL PROGRAMMING (GP)

Goal programming is a variation of linear programming that can be

used for problems that involve multiple objectives

In linear and integer programming methods the objective function is

measured in one dimension only

It is not possible for LP to have multiple goals unless they are all

measured in the same units, and this is a highly unusual situation

A variation of linear programming that allows multiple objectives (goals)

— soft (goal) constraints or a combination of soft and hard (non goal)

constraints—that can deviate, allowing for tradeoffs in achieving

satisficing rather than only optimal solutions.

Charnes, Cooper and Ferugson in 1955

To solve Multi criteria dilemma – due to constraints in LP

First application- design & placement of antenna in 2nd stage

of Saturn V ( which was used to launch Apollo Space capsule).

Assumptions

Similar to LP:

Non-negative variables

Conditions of certainty

Variables are independent

Limited resources

Deterministic

Differences LP and GP

Linear programming LP Goal programming GP

Goal and objectives One primary- to be maximized or minimized

All objectives are ranked each with a target

Goals or constraints Inflexible, no deviations are allowed

Flexible, deviations are acceptable, constraints can be relaxed

Objective functions Maximize (minimize) the value of the primary goal

Minimize the sum of the undesirable deviations (weighted by their relative importance)

Theory Optimization Satisfaction

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 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

v Represents overachieving level of the goal

u Represents underachieving level of the goal

Economic (hard) Constraints

Stated as <=, >=, or =

Linear (stated in terms of decision variables)

Example: 3x + 2y <= 50 hours

Goal (soft) Constraints

General form of goal constraint:

+ u - v =

Goal Programming Constraints

Decision

Variables

Desired Goal

Level

• No pre assigned priority

• All the goals are of roughly comparable importance

Decision makers are merely required to rank deviations from the various

goals in their order of importance

Weights are assigned to the goals to distinguish the importance of various goals

PRE-EMPTIVE GOAL PROGRAMMING

Goal Programming Example

Info soft is a growth oriented firm which establishes monthly

performance goals for its sales force

Info soft 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


Info soft 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/hard Constraints

Goal/soft 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

v1= the number of current clients visited in excess of the goal of 200

u1= the number of current clients visited less than the goal of 200

v2= the number of new clients visited in excess of the goal of 120

u2= 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

v1, u1, v2, u2 => 0

Goal Constraints:

Current Clients: X1 + u1 – v1= 200

New Clients: X2 + u2 - v2= 120Must be =


Objective Function:

Minimize Weighted Deviations

Minimize u1, u2

Model formulation

Complete formulation:

Minimize u1, u2

Subject to:

2X1 + 3X2 <= 640

X1 + u1 – v1= 200

X2 + u2 – v2= 120

X1, X2 => 0

u1,v1,u2,v2 => 0

Graphical Solution

Graph constraint:

2X1 + 3X2 = 640

If X1 = 0, X2 = 213

If X2 = 0, X1 = 320

Plot points (0, 213) and (320, 0)

Graphical Solution

0 50 100 150 200

50

100

150

200

X2

250 300 350

(0,213)

(320,0)X1


Graph deviation lines

X1 + u1 – v1= 200 (Goal 1)

X2 + u2 – v2= 120 (Goal 2)

Plot lines for X1 = 200, X2 = 120


0 50 100 150 200

50

100

150

200

X1

X2 Goal 1

u1 v1

(140,120)

(200,80)

250 300 350

(0,213)

(320,0)

Goal 2 v2

u2

Solving Graphical Goal Programming

Want to Minimize u1 + u2

So we evaluate each of the candidate solution points:

For point (140, 120)

u1 = 60 and u2= 0

Z = 60 + 0 = 60

For point (200, 80)

u1 = 0 and u2 = 40

Z = 0 + 40 = 40

Optimal Point

Contact at least 200 current

clients

Contact at least 120 new

clients

Maximize Z = $40x1 + 50x2

subject to:1x1 + 2x2 40 hours of labor4x2 + 3x2 120 pounds of clayx1, x2 0

Where: x1 = number of bowls producedx2 = number of mugs produced

P1 Does not want to use fewer than 40 hours of labor per day.

P2 Would like to achieve a satisfactory profit level of $1,600 per day.

P3 Prefers not to keep more than 120 pounds of clay on hand each day.

P4 Would like to minimize the amount of overtime.

Priority Goal

A Plot of a Goal Constraint

V1

U1

The first-priority

goal:

minimize

U1

(d1- )

The 2nd -priority

goal:

minimize

U2

(d2- )

The 3rd -priority

goal:

minimize

V3

(d3+ )

The 4th -priority

goal:

minimize

V1

(d1+ )

A manufacturing firm produces two types of products A and B .The

unit profit from the product A is 100$ and that of product B is 50$. The

goal of the firm is to earn a total profit of exactly $700 in the next

week. The manager in addition to the profit goal of 700$ also wishes

to achieve sales volume for products A and B close to 5 and 4

respectively.

Formulate this problem as a GP Model.

1. Identify the decision variables

2. Identify the constraints and determine which ones are goal constraints

3. Formulate the non-goal (hard) economic constraints (if any)

4. Formulate the goal (soft) constraints

5. Formulate the objective function

6. Add the non-negativity requirement statement

A company manufactures 3 products: X1, X2 and X3. The material and labour requirements per unit are:

The manager has listed the following objectives in order of priority:

1. Minimize overtime in the assembly department

2. Minimize undertime in the assembly department

3. Minimize both undertime and overtime in the packaging department

Find out optimum mix of X1, X2, and X3.

Product A B C Availability

Material (lb/ unit) 2 4 3 600

Assembly time

(min./ unit)

9 8 7 900

Packing time (min

/ unit)

1 2 3 300

The constraints are:

2 X1+ 4 X2+ 3 X3 <= 600 non-goal constraint (hard)

9 X1 + 8 X2 + 7 X3 + u1 –v1 = 900 goal constraint (soft)

1 X1 + 2 X2 + 3 X3 + u2 –v2 = 300 goal constraint (soft)

All variables >= 0

The objective is based on listing the deviation variables based on priorities

Minimize P1v1, P2u1, P3 (u2+ v2)

Minimize P1u1, P2u2, P3 u3


5 X1+ 3 X2<= 150 (A) non-goal constraint (hard)

2 X1 + 5 X2 + u1 –v1 = 100 (1) goal constraint (soft)


X1 + u3 –v3 = 40 (3) goal constraint (soft)

All variables >= 0

A Plot of a Goal Constraint

V1

U1

Designating Priority and Direction

Plot of the Hard Constraint and the Feasible Solution Space

5 X1+ 3 X2 <= 150

The Acceptable Region after Adding the First Goal Constraint

Minimize P1u1

The Second Goal Is Added to the Graph

Minimize P2u2,

The Third Goal Is Added, but It Doesn’t Change the Solution

Minimize P3 u3

•Minimize P1v1, P2u2, P3 v3

•Subject to;

Plotting the Acceptable Region

Robinson chemical manufacturing company produces two types of chemical compounds and its called as compound 100 and

compound 200. Each unit of compound 100 uses 5 pounds of mixing material. Each unit of compound 200 uses 3 pounds of

mixing material. There are 150 pounds of mixing material available per day. It takes 2 hours labour time to make one unit of

compound 100, while each unit of compound 200 uses 5 hours of labour there are 100 hours of labour available per shift and the

company operates one shift per day. The chemicals (materials used to mix the compounds) have to be mixed and cured using

mixer 5000. it takes 3 hours to mix and cure one unit of compound 100 and 3.6 hours to mix and cure a unit of compound 200.

there are 180 machine hours per day available for mixer 5000. Daily demand estimated to be 40 units for compound 100 and 50

units for compound 200. the company identified the following four priority goals for this problem.

1. Minimize being under the labour hours constraint

2. Minimize being under the machine hours constraint

3. Minimize being under the demand constraint for compound 200

4. Minimize being under the demand constraint for compound 100

5. Minimize labour overtime

Minimize P1u1, P2u2,P3 u4 ,P4 u3,P5 v1


5 X1+ 3 X2<= 150 (A) non-goal constraint (hard), material

2 X1 + 5 X2 + u1 –v1 = 100 (1) goal constraint (soft), Labour

3 X1 + 3.6 X2 + u2 –v2 = 180 (2) goal constraint (soft), machine

X1 + u3 –v3 = 40 (3) goal constraint (soft), demand X1

X2 + u4 –v4 = 50 (4) goal constraint (soft), demand X2

All variables >= 0

Goal Weight

Minimize being under the labour hours constraint35

Minimize being under the machine hours constraint25

Minimize being under the demand constraint for compound 20020

Minimize being under the demand constraint for compound 10015

Minimize labour overtime 5

The manager assigns weight to the following goals. he used 100 points approach to

assign weights.

Minimize weighted deviation variables

35 u1 + 25 u2+ 20u4 + 15u3+ 5v1


5 X1+ 3 X2<= 150 (A) non-goal constraint (hard)


3 X1 + 3.6 X2 + u2 –v2 = 180 (2) goal constraint (soft)



All variables >= 0

simplicity and ease of use

applications in many and diverse fields

handle relatively large numbers of variables, constraints and objectives

A debated weakness is the ability of goal programming to produce solutions that are not Pareto efficient.

This violates a fundamental concept of decision theory, that is no rational decision maker will knowingly choose a solution that is not Pareto efficient.

For a given GP problem, assigning different priority ranks or priority weights among different

goals will result in different solutions. So, GP is viewed as a heuristic approach to multi

criteria Linear problems.

There is no sensitivity analysis information provided along computer solution printouts. To

answer managerial what-if questions:

Trail-and-error (interactive resolving) methods are usually used to check whether there is

an impact on the solution values when:

Changing in priority ranking of goal settings,

Changing in the target value of goals, and

Changing in the weights assigned to each goal deviation.

Introduction to management sciences with spread sheets by Stevenson and ozgurTata McGraw-Hill

Google books link:

http://books.google.co.in/books?id=5PeNDuM-pnQC&printsec=frontcover&source=gbs_ge_summary_r&cad=0#v=onepage&q&f=false

Introduction to Management Science by Bernard W. Taylor, pearson publications

Pdf copy download link:

https://www.academia.edu/7219878/Introduction_to_Management_Science_11th_Edition-_Bernard_W_Taylor_III

http://books.google.co.in/books?id=5PeNDuM-pnQC&printsec=frontcover&source=gbs_ge_summary_r&cad=0#v=onepage&q&f=false

https://www.academia.edu/7219878/Introduction_to_Management_Science_11th_Edition-_Bernard_W_Taylor_III

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Goal Programming