1. Goal Programming and Multiple Objective Optimization
December 08, 2014

2. Dave Powell, Elon University, dpowell2@elon.edu
Goal Programming
Choose a target value for each objective (dealing in criterion space)
Minimize deviation from target and objective
Create a constraint for each objective
If minimize a goal
Minimize PositiveDeviation from Target
subject to: F(x) – PositiveDeviation <= target
PostiveDeviation >=0
If maximize
Minimize NegativeDeviation from Target
subject to: F(x) + NegativeDeviation >= target
If equal to target
Minimize PositiveDeviation + Negative Deviation from target
Subject to: F(x) – Positive Deviation + NegativeDeviation = target
Source:
Dave Powell, Elon University,

3. Goal Programming (GP) In LP problems there are;
Hard constraints: constraints that cannot be violated
Examples
There are 1,566 labor hours available.
There is $850,00 available for projects.
Soft constraints: represent goals or targets we’d like to achieve
You have a maximum price in mind when buying a car (this is your “goal” or target price)
If you can’t buy the car for this price you’ll likely find a way to spend more
Source: A Practical Introduction to Management Science 5th edition. Cliff T. Ragsdale

4. Goal Programming Example:
Mr. X wants to expand the convention center at his hotel
The types of conference rooms being considered are:
Size (sq. ft.)
Unit Cost ($)
Small
400
18,000
Medium
750
33,000
Large
1,050
45,150
I - Mr. X would like to add
5 small,
10 medium and
15 large conference rooms
II - He also wants the
total expansion = 25,000 square feet
the cost limited to $1,000,000
Source: A Practical Introduction to Management Science 5th edition. Cliff T. Ragsdale

5. Decision Variables X1 = number of small rooms to add
X2 = number of medium rooms to add
X3 = number of large rooms to add
Source: A Practical Introduction to Management Science 5th edition. Cliff T. Ragsdale

6. Defining the Goals
Goal 1: The expansion should include approximately 5 small conference rooms
Goal 2: The expansion should include approximately 10 medium conference rooms
Goal 3: The expansion should include approximately 15 large conference rooms
Goal 4: The expansion should consist of approximately 25,000 square feet
Goal 5: The expansion should approximately cost $1,000,000
Source: A Practical Introduction to Management Science 5th edition. Cliff T. Ragsdale

7. Defining the Goal Constraints-I
Small Rooms
Medium Rooms
Large Rooms
Source: A Practical Introduction to Management Science 5th edition. Cliff T. Ragsdale
where

8. Defining the Goal Constraints-II
Total Expansion
Total Cost (in $1,000s)
where
Source: A Practical Introduction to Management Science 5th edition. Cliff T. Ragsdale

9. GP Objective Functions
There are numerous objective functions we could formulate for a GP problem.
Minimize the sum of the deviations:
Problem: The deviations measure different things, so what does this objective represent?
Source: A Practical Introduction to Management Science 5th edition. Cliff T. Ragsdale

10. GP Objective Functions (cont’d)
Minimize the sum of percentage deviations
MIN
where ti represents the target value of goal i
Problem: Suppose the first goal is underachieved by 1 small room and the fifth goal is overachieved by $20,000
We underachieve goal 1 by 1/5=20%
We overachieve goal 5 by 20,000/1,000,000= 2%
This implies being $20,000 over budget is just as undesirable as having one fewer small rooms.
Source: A Practical Introduction to Management Science 5th edition. Cliff T. Ragsdale
Is this true? Only the decision maker can say for sure.

11. GP Objective Functions (cont’d)
Weights can be used in the previous objectives to allow the decision maker indicate
desirable vs. undesirable deviations
the relative importance of various goals
Minimize the weighted sum of deviations
MIN
Source: A Practical Introduction to Management Science 5th edition. Cliff T. Ragsdale
Minimize the weighted sum of % deviations
MIN

12. Defining the Objective
Assume
It is undesirable to underachieve any of the first three room goals
It is undesirable to overachieve or underachieve the 25,000 square feet expansion goal
It is undesirable to overachieve the $1,000,000 total cost goal
Source: A Practical Introduction to Management Science 5th edition. Cliff T. Ragsdale
Initially, assume all the above weights equal 1

13. Comments About GP
GP involves making trade-offs among the goals until the most satisfying solution is found
GP objective function values should not be compared because the weights are changed in each iteration. Compare the solutions!
An arbitrarily large weight will effectively change a soft constraint to a hard constraint
Source: A Practical Introduction to Management Science 5th edition. Cliff T. Ragsdale

14. The MiniMax Objective
Can be used to minimize the maximum deviation from any goal
MIN: Q
etc...
Source: A Practical Introduction to Management Science 5th edition. Cliff T. Ragsdale

15. Summary of Goal Programming
1. Identify the decision variables in the problem
2. Identify any hard constraints in the problem and formulate them in the usual way
3. State the goals of the problem along with their target values
4. Create constraints using the decision variables that would achieve the goals exactly
5. Transform the above constraints into goal constraints by including deviational variables
6. Determine which deviational variables represent undesirable deviations from the goals
7. Formulate an objective that penalizes the undesirable deviations
8. Identify appropriate weights for the objective
9. Solve the problem
10. Inspect the solution to the problem. If the solution is unacceptable, return to step 8 and revise the weights as needed
Source: A Practical Introduction to Management Science 5th edition. Cliff T. Ragsdale

16. Multiple Objective Optimization

17. Multiple Objective Linear Programming (MOLP)
An MOLP problem is an LP problem with more than one objective function
Can be considered as special type of GP

18. Multiple Objective Problems
Problems with more than 1 objective
Most of the times objectives are conflicting
Example: Operating strategy for a large multipurpose reservoir
Water Supply: Maximize storage
Flood mitigation: Minimize storage
Recreation: keep water level constant

19. Quiz?
Write 2 problems with conflicting objectives from any subject area.

20. An Example: Cost Comfort Reliability Power Cuore Subaru Wagon Jaguar
Purchase a Car
Cost
Comfort
Reliability
Power
Cuore
Subaru Wagon
Jaguar
Shezore
Source:

21. Introduction
Multi-objective programming deals with optimization problems with two or more objective functions
Differs from the classical (single-objective) optimization problem only in the expression of their respective objective functions
Instead of seeking an optimal or best overall solution, the goal of multi-objective analysis is to quantify the degree of conflict, or tradeoff among objectives
Source: Revelle

22. Trade-offs Between Objectives
Profit C B A
Toxic Waste

23. Concept in conflicting objectives
In a multiple objective problems the solution optimizing one objective may not be optimizing other objectives
In real life problems, usually there is a very large degree of conflict between objectives
Many times the units of measure for different objective functions may be quite different
such objective functions are called noncommensurate

24. Concept in conflicting objectives..conti
For conflicting objectives, the concept of optimality may be inappropriate
a strategy that is optimal with respect to one objective may likely be clearly inferior for another
Concept of noninferiority is used to measure solutions against multiple, conflicting, and even noncommensurate objectives

25. Noninferior solutions
Try to find the set of solutions for which you can demonstrate that no better solutions exist
This best available set of solutions is referred to as the set of noninferior solutions
From this set decision makers choose their solution

26. A solution to a problem having multiple and conflicting objectives is noninferior if there exists no other feasible solution with better performance with respect to any one objective, without having worse performance in at least one other objective
Note:
The solutions that optimize the individual objective functions will be noninferior (unless there are alternate optima)
All interior points within feasible region must be inferior
Noninferiority is even called nondominance by some mathematical programmers, efficiency by statisticians and economists, and Pareto optimality by welfare economists
for any multiobjective optimization model, the solution that optimizes any single objective function is always noninferior, unless there are
alternate optima

27. December 10, 2014

28. Non Inferiority
“A solution to MOP is non-inferior if there exists no other solution that will yield an improvement in one objective without causing a degradation in at least one other objective.”
Maximize
Power
Source:
Environmental Quality

29. Minimum Instream Fish Flow, in 1000 cfs
Correspondence exists between feasible region in decision space and the feasible region in objective space.
Size of Installed
Turbine
Minimum Instream Fish Flow, in 1000 cfs 1 2 3 4 5 6 5 10 15 20 25 30 35
Power Production
Environmental Quality
Source:

30. Exercise
Source:

31. Source: http://civil. colorado

32. Air pollution concerns and the results of an air monitoring program have identified the release of contaminated dust during the blending process; the binder used in manufacturing both HYDIT and FILIT attaches to these duct particles and is thereby released to the environment during production. Lab tests indicated that the release of this pollutant from the plant amounts to 500 mg/ton of HYDIT produced, and 200 mg/ton of FILIT produced.
A second objective is now introduced to minimize plant emissions

33. Example Problem: What is optimal production strategy?
Objective 1: Maximize total weekly revenue ($)
Maximize Z1 = 140X X2
Objective 2: Minimize emissions (milligram)
Minimize Z2 = 500X X2
The two objective functions are related through the same set of decision variables

34. x1=6, x2=4
Z1=1480
Source:
x1=0, x2=0
Z2=0

35. Source: http://civil. colorado

36. Source: http://civil. colorado

37. Noninferior solutions
The determination of noninferiority gets increasingly complicated as the problem grows in size, in both number of basic feasible extreme point solutions and number of objective functions
Most real engineering problems in the public sector have hundreds of thousands of feasible extreme points and may have tens of objective functions
Goal is to identify the Pareto optimal set and understand the tradeoffs between solutions
Source:

38. Another example with 8 possible solutions and 3 objectives
Source:

39. Source: Revelle

40. Decision Space vs. Objective Space
Because we have limited our example problem to not more than 2 objectives, we can map the feasible region in decision space to a corresponding feasible region in objective space
The shape of the feasible region in decision space depends on the constraint set for a particular problem
The shape of the feasible region in objective space depends on the objective functions, which serve as “mapping functions” for a particular set of objectives

41. Source: Revelle

42. The goal of such an analysis is thus to identify all solutions that are noninferior: the set of solutions for which there does not exist another solution that would always be preferable to any of those solutions

43. Method to find the noninferior set
Weighting Method

44. Introduction
Different weights are used to generate different Pareto optimal solutions
DM selects the most satisfactory to him

45. The weighting method can be accomplished as follows:
Solve ‘p’ linear programs, each having a different objective function
Each of these solutions is a noninferior solution for the ‘p’ objective function problem provided …………..
Combine all objective functions into a single-objective function by multiplying each objective function by a weight and adding them together such that;
Maximize Z = [Z1, Z2, ……Zp] becomes
Maximize Z (w1, w2, ….wp)= [w1 Z1, w2 Z2, …… wp Zp]
This objective function is often referred to as the grand objective.
provided that alternate optima do not exist at that solution. If alternate optima are indicated, at least one of the optimal basic feasible extreme points will be noninferior (it is possible that more than one will be noninferior, but likely that some will be dominated).

46. Conti….
Solve a series of linear programs using the grand objective while systematically varying the weights on the individual objectives
Each of these solutions will be a noninferior solution for the multi-objective problem
Note: For minimization objectives, one can multiply that objective function by —1 to change its sense to a maximization

47. Weighting method to solve two-objective problem - Example
Grand objective function

48. Revelle

50. Dealing with Alternate Optima When Using the Weighting Method
Read reference document (Chapter 5 - Revelle)

51. Method to find the noninferior set
Constraint Method

52. Constraint Method
Solve ‘p’ individual models to identify the solution that optimizes each
Selecte (arbitrarily) one objective function to be optimized and include other objective functions in the constraint set with right-hand sides set so as to restrain the value of the objective function that was selected for optimization and solve
Noninferior set in the objective space comprises the solutions to the modified problems

53. Example Original Problem Solve for Z2
solution x1 = 3, x2 = 6, and Z*= 9

58. Analytical Hierarchy Method
Method widely introduced by Thomas L. Saaty
Requires hierarchical organization of problem
Performed by comparing activities at different levels
Uses pair-wise comparisons
Provides techniques to incorporate subjective information
Source:

59. The Scale With respect to any criteria (e.g. cost) compare
alternatives A and B
Score Meaning
1 A and B are equally important
3 A is weakly more important than B
5 A is more important than B
7 A is very strongly more important than B
9 A is absolutely more important than B
1/9 A is absolutely less important than B
1/7 A is very strongly less important than B
1/5 A is less important than B
1/3 A is weakly less important than B

60. References
A Practical Introduction to Management Science 5th edition. Cliff T. Ragsdale