Multi-Criteria Decision Making and Optimization

Ankur SinhaAnkur.Sinha@aalto.fi

Department of Information and Service EconomyAalto University School of Economics

Ankur Sinha

- Constraint Method If followin m objective optimization problem is to be solved

Minimize f(x) = (f1(x),f2(x), .. ,fm(x) )

Subject to x S Constrain all the objectives except one, say Choose a relevant vector Solve the following single objective optimization problem

Minimize f(x)

Subject to fi(x) i, i {1,..m}, i x S

Each vector leads to a solution on the Pareto-optimal frontier

Ankur Sinha

- Constraint MethodFor a two objective minimization problemMin f(x) = (f1(x),f2(x) )

Subject to x S

We solve the following single objective mathematical program for different values of 1Min f2(x)

Subject to f1(x) 1x S

Ankur Sinha

- Constraint Method

Disadvantage: Requires relevant vectors Disadvantage: Leads to non-uniform Pareto-optimal

solutions Advantage: Guarantees Pareto-optimal solution on the

frontier Advantage: Does not suffer even if the Pareto-frontier is

non-convex (minimization problems)

Ankur Sinha

ExampleConsider the following multi-objective problemMin f1(x,y) = x2 + y2

Min f2(x,y) = (x-2)2 + (y-2)2

x,y [0,2]xx22

xx11

ff22

ff11

SS

0 2

2(1.5,1.5)

(4.5,0.5)

2 4

4

2

Ankur Sinha

Example

xx22

xx11

SS

0 2

2

ff22

ff114

8

4

ff(S)(S)

8

Ankur Sinha

Example

Min f2(x,y) = (x-2)2 + (y-2)2

Subject to

f1(x,y) 6

Solution

Optimal (x,y) : (1.73,1.73) Optimal (f1,f2): (6,0.14)

ff22

ff114

8

4

862

Feas

ible

Reg

ion

for

Eps

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Ankur Sinha

Example

Min f2(x,y) = (x-2)2 + (y-2)2

Subject to

f1(x,y) 4

Solution

Optimal (x,y) : (1.41,1.41) Optimal (f1,f2): (4,0.69)

ff22

ff114

8

4

862

Feas

ible

Reg

ion

for

Eps

ilon

Con

stra

int P

robl

em

Ankur Sinha

Example

Min f2(x,y) = (x-2)2 + (y-2)2

Subject to

f1(x,y) 2

Solution

Optimal (x,y) : (1,1) Optimal (f1,f2): (2,2)

ff22

ff114

8

4

862Fe

asib

le R

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n fo

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onst

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t Pro

blem

Ankur Sinha

Example

Min f2(x,y) = (x-2)2 + (y-2)2

Subject to

f1(x,y) 1

Solution

Optimal (x,y) : (0.71,0.71) Optimal (f1,f2): (1,3.32)

ff22

ff114

8

4

8621

Feasible Region for Epsilon Constraint Problem

Ankur Sinha

ff22

ff114

8

4ff(S)(S)

862

Example

1

Choose other values for to get more points on the frontier

= 1

= 2

= 4

= 6

Ankur Sinha

Goal Programming It is a linear programming problem which satisfies

multiple goals at the same time

Multiple goals are prioritized and weighted to account for the decision maker's requirements

Minimizes sum of weighted deviations from the target values It is ALWAYS the objective for Goal Programming

Ankur Sinha

Assume Let

gi: goal to be achieved in criteria iI: {index set of objectives for which under-

achievement is undesirable}M: {index set of objectives for which over-achievement

is undesirable}K: {index set of objectives for which both under-

achievement and over-achievement are undesirable}

=

=

n

jjiji xcxz

1)( i=1,,p

Goal Programming

Ankur Sinha

'

1

( )i i m m k k kki I m M k K

n

ij j i i ij

z w d w d w d w d

c x g d d

x X

+ +

+

=

= + + +

= +

+

=

+= iiinj

jij ddgxc1

Min

wi is a constant weight for the deviation which is provided by the decision maker

Underachievement Variable

Overachievement Variable

Goal Objective

Goal Constraints

Goal Programming

Ankur Sinha

Goal Programming Steps

Define decision variablesDefine deviation variables for each goalFormulate constraint equations

Economic constraintsGoal constraints

Formulate objective function

Ankur Sinha

Decision variables are the unknown variables in the optimization problemDeviation variables represent overachieving or underachieving each goal

d+ Represents overachieving level of the goal d- Represents underachieving level of the goal

Goal Programming Variables

Ankur Sinha

Economic ConstraintsStated as , , or = Linear (stated in terms of decision variables)Example: 3x + 2y 50 hours

Goal ConstraintsGeneral form of goal constraint:

Goal Programming Constraints

CriteriaGoal + d+ - d- =

Ankur Sinha

Goal Programming Example

Fincom is a growth oriented firm which establishes monthly performance goals for its sales forceFincom determines that the sales force has a maximum available hours per month for visits of 640 hoursFurther, it is estimated that each visit to a potential new client requires 3 hours and each visit to a current client requires 2 hours

Ankur Sinha

Fincom establishes two goals for the coming month:

Contact at least 200 current clientsContact at least 120 new clients

Overachieving either goal will not be penalized

Goal Programming Example

Ankur Sinha

Steps Required: Define the decision variables Define the goals and deviation variables Formulate the goal programming model

parameters: Economic Constraints Goal Constraints Objective Function

Goal Programming Example

Ankur Sinha

Step 1: Define the decision variables:X1 = the number of current clients visitedX2 = the number of new clients visited

Step 2: Define the goals:Goal 1 Contact 200 current clientsGoal 2 Contact 120 new clients

Goal Programming Example

Ankur Sinha

Step 3: Define the deviation variablesd1+ = the number of current clients visited in excess of the goal of 200d1- = the number of current clients visited less than the goal of 200d2+ = the number of new clients visited in excess of the goal of 120d2- = the number of new clients visited less than the goal of 120

Goal Programming Example

Ankur Sinha

Formulate the GP Model:Economic Constraints:

2X1 + 3X2 640 X1, X2 0 d1+, d1-, d2+, d2- 0

Goal Constraints: Current Clients: X1 + d1- - d1+ = 200 New Clients: X2 + d2- - d2+ = 120

Must be =

Goal Programming Example

Ankur Sinha

Objective Function:Minimize Weighted DeviationsMinimize Z = d1- + d2-

Goal Programming Example

Ankur Sinha

Complete formulation:Minimize Z = d1- + d2-

Subject to:2X1 + 3X2 640X1 + d1- - d1+ = 200X2 + d2- - d2+ = 120X1, X2 0d1+, d1-, d2+, d2- 0

Goal Programming Example

Ankur Sinha

Economic constraint:2X1 + 3X2 = 640

If X1 = 0, X2 = 213If X2 = 0, X1 = 320

Plot points (0, 213) and (320, 0)

Goal Programming Example

Ankur Sinha

00 5050 100100 150150 200200

5050

100100

150150

200200

XX22

250250 300300 350350

(0,213)(0,213)

(320,0)(320,0)XX11

Goal Programming Example

Ankur Sinha

Graph deviation linesX1 + d1- - d1+ = 200 (Goal 1)X2 + d2- - d2+ = 120 (Goal 2)

Plot lines for X1 = 200, X2 = 120

Goal Programming Example

Ankur Sinha

Goal Programming Example

00 5050 100100 150150 200200

5050

100100

150150

200200

XX11

XX22 Goal 1Goal 1

dd11-- dd11++

Goal 2Goal 2dd22++

dd22--(140,120)(140,120)

(200,80)(200,80)

2X2X11 + 3X + 3X

22 < = 640

< = 640

250250 300300 350350

(0,213)(0,213)

(320,0)(320,0)

Ankur Sinha

Graphical Solution

Want to Minimize d1- + d2-

So we evaluate each of the candidate solution points:

For point (140, 120)d1- = 60 and d2- = 0

Z = 60 + 0 = 60

For point (200, 80)d1- = 0 and d2- = 40

Z = 0 + 40 = 40

Optimal Point

Contact at least 200 current clientsContact at least 120 new clients

Ankur Sinha

Goal Programming SolutionX1 = 200 Goal 1 achievedX2 = 80 Goal 2 not achievedd1+ = 0 d2+ = 0d1- = 0 d2- = 40

Z = 40

Ankur Sinha

Weighted Distance Minimization

The weighted L distance between two vectors x and y is defined as:

=

==

=

=

fTchebychef

Euclideanrrectilinea

yxwLp

jjjj

21

(/1

1

Ankur Sinha

Tchebycheff

Euclidean

Rectilinear

Weighted Distance Minimization

Ankur Sinha

2 4 6 8

2

4

6

8A

B

C

DE

Ideal Point(8,8)

Weighted Distance Minimization

Let weights be 1

Consider a set of alternatives in the objective space

Ankur Sinha

Alt Xi1 Xi2 L1 L2 LA 1 8 7 7 7

B 2 6 8 6.32 6

C 3 4 9 6.40 5*

D 4 3 9 6.40 5*

E 8 2 6* 6* 6

Weighted Distance MinimizationPoint closest to the ideal point is chosen based on L for unit weights

Find the closest solutions based on L1,L2 and L3 if weights are (1,2)

Ankur Sinha

Thank You!

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