γλώσσες
Σελίδες
Νομικός
Multi-Criteria Decision Making and Optimization
Ankur [email protected]
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
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) 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
egio
n fo
r E
psilo
n C
onst
rain
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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