goal programming
TRANSCRIPT
GOAL PROGRAMMING
sonia
• 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.
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
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
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.
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
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
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.
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
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.
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
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)
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…
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
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.
• 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
• 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
• 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
• 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
• 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
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
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