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GOAL PROGRAMMING sonia

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Page 1: Goal Programming

GOAL PROGRAMMING

sonia

Page 2: Goal Programming

• Goal programming is a branch of multi objective optimization, which in turn is a branch of multi-criteria decision analysis (MCDA), also known as multiple-criteria decision making (MCDM).

• This is an optimization programme. It can be thought of as an extension or generalisation of linear programming to handle multiple, normally conflicting objective measures.

• Each of these measures is given a goal or target value to be achieved. Unwanted deviations from this set of target values are then minimised in an achievement function.

Page 3: Goal Programming

Goal programming is used to perform three types of analysis:

• Determine the required resources to achieve a desired set of objectives.

• Determine the degree of attainment of the goals with the available resources.

• Providing the best satisfying solution under a varying amount of resources and priorities of the goals.

Page 4: Goal Programming

• There are two types of goal programming models:– Nonpreemtive goal programming – no goal is pre-determined to dominate any other goal.– Preemtive goal programming – goals are assigned different priority levels. Level 1 goal dominates level 2 goal, and so on.

Page 5: Goal Programming

Strengths and weaknesses• A major strength of goal programming is its simplicity

and ease of use.• This accounts for the large number of goal

programming applications in many and diverse fields. • Goal programming can hence 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.

• The setting of appropriate weights in the goal programming model is another area that has caused debate,

Page 6: Goal Programming

Goal Programming (vs. LP)

Multiple Goals (instead of one goal) “Satisfices” (instead of optimize)

Coming as close as possible to reaching the goal

Objective function is the main difference Deviational Variables Minimized (instead of

maximizing profit or minimizing cost of LP) Once the goal programming is formulated,

we can solved it the same as a LP minimization problem

Page 7: Goal Programming

PROBLEM 1 The Harrison Electric Company, located in Chicago’s Old Town

Area, produces two products popular with home renovators: Table fans Ceiling Fans

Both the table fans & ceiling fans requires a two step production process involving wiring & assembly.

It takes about 2 hours to wire each TF & 3 hour to wire a Ceiling fan Assembly of the TF and CF requires 6 & 5 hours respectively.

Page 8: Goal Programming

IN CONTINUE… The production capacity is such that only 12 hours of

wiring time and 30 hours of assembly time are available.

If each TF produced nets the firm $7 and each C.Fan $6

Formulate the problem as an LP to maximize the ….? X1 = Numbers of TF produced X2 = Numbers of C.Fans produced

Page 9: Goal Programming

LP Formulation Maximize the Profit

Z= $7x1 + $6x2

Subject to 2x1 + 3x2 =< 12 (Wiring Hours) 6 x1 + 5 x2 =< 30 (Assembly Hours) x1 ,x2 >= 0

Page 10: Goal Programming

Problem In Continue.. Lets assume that the firm is moving to a new location

during a particular production period and feels that maximizing the profit is not an realistic goal.

Management sets a profit level of $30. We now have a goal programming problem in which

want to find the right production mix that achieve that profit level.

Page 11: Goal Programming

d1- Underachievement of the profit target

d1+

Overachievement of the profit target Minimize the under or overachievement of the profit

target d1- + d1+

Subject to $7x1 + $6 x2 + d1+- d1+=$30 (Profit goal) 2 x1 + 3 x2 ≤ 12 (Wiring Hours Constraint) 6 x1+ 5 x2 ≤ 30 (Assembly Hours Constraint) x1, x2d1ˉ, d1+ ≥ 0

 

Page 12: Goal Programming

Extension to Equally Important multiple Goals

Lets now look at the situation in which Harrison’s management wants to achieve several goals, each equal in priority.

Goal 1: To produce profit of $30 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.

Page 13: Goal Programming

Deviational variables… d1ˉ = Underachievement of the profit target d1+ = Overachievement of the profit target d2ˉ = Idle time in the wiring department (Underutilization) d2+ = Overtime in the wiring department (Overutilization) d3ˉ = Idle time in the assembly department

(Underutilization) d3+ = Overtime in the assembly department

(Overutilization) d4ˉ = Underachievement of the ceiling fan goal d4+ = Overachievement of the ceiling fan goal

Page 14: Goal Programming

LP Formulation The new objective functions &

constraints are Minimize total deviations =

d1ˉ + d2ˉ +d3+ + d4ˉ Subject to

7x1 + 6x2 + d1ˉ - d1+ = (Profit Constraint) 2x1 + 3x2 + d2ˉ - d2+ = (Wiring Hours Con) 6x1 + 5x2 + d3ˉ - d3+ = (Assembly Con) X2 + d4ˉ - d4+ = 7 (Ceiling Fan Cons)

Page 15: Goal Programming

Ranking Goals with Priority Levels

In most goal programming problems, one goal will be more important than another, which in turn will be more important than a third.

The idea is that a goal can be ranked with respect to their importance in management’s eye.

Lower order goals are considered only after higher order goals are met.

Priorities (PiS) are assigned to each deviational variables, with the ranking that P1 is the most important goal, P2 the next most important, then P3, & so on…

Page 16: Goal Programming

Priority Table of HarrisonGOAL PRIORITY

Reach a profit as much above $30 as possible P1

Fully use wiring department hours available P2

Avoid assembly department overtime P3

Purchase at least seven ceiling fans P4

Minimize Total Deviation = p1d1ˉ + p2d2ˉ + p3d3+ + p4d4ˉ

Page 17: Goal Programming

Areas of GP

• Business organisation• Govt Agencies• Non profit institutions

Page 18: Goal Programming

APPLICATIONS1. Marketing applications:• Media planning- so that it cover max consumer

and min budget.• Marketing logistics- so that the cost should be

min and time should also min and profit is max• Product mix decisions- what should be the

product mix so that profit is max and cost is min.

Page 19: Goal Programming

• A company is considering three forms of advertising.

• Goals– 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.

NONPREEMTIVE GOAL PROGRAMMINGAn Advertisement Example

Cost per Ad CustomersTelevision 3000 1000Radio 800 500Newspaper 250 200

Page 20: Goal Programming

• If these were constraints rather than goals we would have:3000X1 + 800X2 + 250X3 £ 25,0001000X1 + 500X2 + 200X3 ³ 30,000

X1 ³ 10• No feasible solution exists that satisfies all

the constraints.• When these constraints are simply goals

they are to be reached as close as possible.

An Advertisement Example

Page 21: Goal Programming

• Detrimental variablesDi- = under achievement of goal Di+ = over achievement of goal

• The goal equations3000X1 + 800X2 + 250X3 + d1- – d1+ = 25,0001000X1 + 500X2 + 200X3 + d2- – d2+ = 30,000

X1 + d3- – d3+ = 10

An Advertisement Example

Page 22: Goal Programming

• 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

Page 23: Goal Programming

• It is assumed that no advantage is gained by overachieving a goal.

Minimize 1E1 + 5U2 + 100U3

s.t.3000X1 + 800X2 + 250X3 + U1 – E1 = 25,0001000X1 + 500X2 + 200X3 + U2 – E2 = 30,000

X1 + U3 – E3 = 10All variables are non-negative.

The goal programming model

Page 24: Goal Programming

2. Academic application

• Assigning faculty teachings- so that gap b/w lectures is proper and every teacher get free as well as important lectures

• University admission planning- how many counter should be there so that students don’t have to wait in long lines and also time consume is minimum

Page 25: Goal Programming

3. Finance applications

• Portfolio selection: max return and min risk• Capital budgeting• Financial planning

Page 26: Goal Programming

4. HRD

• Transportation problem of staff-so that everybody reach on time and cost and time will be minimised

• Manpower planning-min number of selected persons and max no of job assigned

Page 27: Goal Programming

5. Public system

• Transportation system• Health care delivery planning