Goal Programming

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Goal programming is a branch of multi objective optimization, which in turn is a branch of multicriteria 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.

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.

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.

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,

Goal Programming (vs. LP)Multiple Goals (instead of one goal) Satisfices (instead of optimize)

Coming as close as possible to reaching the goal Deviational Variables Minimized (instead of maximizing profit or minimizing cost of LP)

Objective function is the main difference

Once the goal programming is formulated, we can solved it the same as a LP minimization problem


The Harrison Electric Company, located in Chicagos 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.


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

LP Formulation

Maximize the Profit

Z= $7x1 + $6x2

Subject to2x1 + 3x2 =< 12 (Wiring Hours) 6 x1 + 5 x2 =< 30 (Assembly Hours) x1 ,x2 >= 0

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.


Underachievement of the profit target 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, x2d1, d1+ 0

Extension to Equally Important multiple Goals

Lets now look at the situation in which Harrisons 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.

Deviational variables

d1 = Underachievement of the profit target d1+ = Overachievement of the profit target d2 = Idle time in the wiring department (Underutilization) d2+ = Overtime in the wiring department (Overutilization) d3 = Idle time in the assembly department (Underutilization) d3+ = Overtime in the assembly department (Overutilization) d4 = Underachievement of the ceiling fan goal d4+ = Overachievement of the ceiling fan goal

LP FormulationThe new objective functions & constraints are Minimize total deviations =

d1 + d2 +d3+ + d4

Subject to7x1 + 6x2 + d1 - d1+ = (Profit Constraint) 2x1 + 3x2 + d2 - d2+ = (Wiring Hours Con) 6x1 + 5x2 + d3 - d3+ = (Assembly Con) X2 + d - d4+ = 7 (Ceiling Fan Cons)4

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 managements 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

Priority Table of HarrisonGOAL Reach a profit as much above $30 as possibleFully use wiring department hours available Avoid assembly department overtime Purchase at least seven ceiling fans


Minimize Total Deviation = p1d1 + p2d2 + p3d3+ + p4d4

Areas of GP Business organisation Govt Agencies Non profit institutions

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.

NONPREEMTIVE GOAL PROGRAMMING An Advertisement Example A company is considering three forms of advertising.Cost per Ad Customers 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

If these were constraints rather than goals we would have:3000X1 + 800X2 + 250X3 25,000 1000X1 + 500X2 + 200X3 30,000 X1 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 Detrimental variablesDi- = under achievement of goal Di+ = over achievement of goal

The goal equations3000X1 + 800X2 + 250X3 + d1- d1+ = 25,000 1000X1 + 500X2 + 200X3 + d2- d2+ = 30,000 X1 + d3- d3+ = 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.

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.

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 dont have to wait in long lines and also time consume is minimum

3. Finance applications Portfolio selection: max return and min risk Capital budgeting Financial planning

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

5. Public system Transportation system Health care delivery planning