Goal Programming and Multiple Objective Optimization December 08, 2014

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: http://civil.colorado.edu/~balajir/CVEN5393/lectures/ Dave Powell, Elon University, dpowell2@elon.edu

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

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

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

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

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

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

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

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.

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

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

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

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

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

Multiple Objective Optimization

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

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

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

An Example: Cost Comfort Reliability Power Cuore Subaru Wagon Jaguar Purchase a Car Cost Comfort Reliability Power Cuore Subaru Wagon Jaguar Shezore Source: http://civil.colorado.edu/~balajir/CVEN5393/lectures/

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

Trade-offs Between Objectives Profit C B A Toxic Waste

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

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

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

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

December 10, 2014

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: http://tinyurl.com/6g4lnjp Environmental Quality

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: http://tinyurl.com/6g4lnjp

Exercise Source: http://tinyurl.com/6g4lnjp

Source: http://civil. colorado

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

Example Problem: What is optimal production strategy? Objective 1: Maximize total weekly revenue ($) Maximize Z1 = 140X1 + 160X2 Objective 2: Minimize emissions (milligram) Minimize Z2 = 500X1 + 200X2 The two objective functions are related through the same set of decision variables

x1=6, x2=4 Z1=1480 Source: http://civil.colorado.edu/~balajir/CVEN5393/lectures/MultiobjectiveOpt-ez.pdf x1=0, x2=0 Z2=0

Source: http://civil. colorado

Source: http://civil. colorado

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: http://civil.colorado.edu/~balajir/CVEN5393/lectures/MultiobjectiveOpt-ez.pdf

Another example with 8 possible solutions and 3 objectives Source: http://civil.colorado.edu/~balajir/CVEN5393/lectures/MultiobjectiveOpt-ez.pdf

Source: Revelle

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

Source: Revelle

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

Method to find the noninferior set Weighting Method

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

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).

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

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

Revelle

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

Method to find the noninferior set Constraint Method

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

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

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: http://civil.colorado.edu/~balajir/CVEN5393/lectures/

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

References A Practical Introduction to Management Science 5th edition. Cliff T. Ragsdale http://civil.colorado.edu/~balajir/CVEN5393/lectures/MultiobjectiveOpt-ez.pdf