1. MO-Problems

2. Suppose that there are some cities numbered 1,2, ..., n,
Multi-Objective TSP
The mathematical model of MOTSP
Suppose that there are some cities numbered 1,2, ..., n,
the distance between two stochastic cities i and j is dis[i][j](i<>j), and
the cost between two stochastic cities is cos[i][j](i<>j),

3. the mathematical model of MOTSP can be expressed as follows:
Where x is a route permutation of n numbers. So function f1 is the summation of the distance from one city to the next whose sequence follows the sequence
in x and function f 2 is the summation of the cost from one city to the next.

4. Coding
chromosome is composed by a series of integer queue, namely, the coding method is based chromosome can not be appeared twice, in order to satisfy the restriction that one city should be passes and be passed only once. For example, there are 10 cities, then an individual
x = [3,4,8,5,9,6,2,7,1,10]
is the route

5. Fitness Calculation
Selection
Crossover
Mutation

6. Quadratic Assignment Problem
The quadratic assignment problem (QAP) is a model for many practical problems like backboard wiring, campus and hospital layout, and scheduling.
Intuitively, QAP can best be described as the problem of assigning a set of facilities to a set of locations with given distances between the locations and given flows between facilities.
The goal then is to place the facilities on locations in such a way that the sum-of-product of flows and distances is minimal.
More formally, given n facilities and n locations, two matrices and , here is the distance between location i and location j and is the flow between facility r and facility s.

7. QAP Example How to assign facilities to locations ? Facilities

8. Quadratic Assignment Problem QAP
ELEMENTS a3
POSITIONS
. . .
. . . an bm
matrix of weights (“flow”) cij
matrix of disctance dij
a1 … aj … an
b1 … bj … bn a1 … ai an
cij b1 … bi bn
dij

9. Quadratic Assignment Problem QAP
A Basic (“flow”) Mathematical Formulation
of the Quadratic Assignment Problem
An assignment of facilities { i | i{1,…,n} } to positions (locations) { p(i) },
where p is permutation of numbers {1,…,n },
a set of all possible permutations is  = { p }.
Let us consider two n by n matrices:
(i)a “flow” (or “utility” )matrix C whose (i,j) element represents the flow between elements (e.g., facilities) i and j, and
(ii)a distance matrix D whose (i,j) ( p(i), p(j) ) element represents
the distance between locations p(i) and p(j).
With these definitions, the QAP can be written as
max ni=1 nj=1 cij dp(i)p(j)
p  

10. Representation !!!!

11. Quadratic Assignment Problem QAP
BASIC BOOKS:
1.P.M. Pardalos, H. Wolkowicz, (Ed.), Quadratic Assignment and
Related Problems. American Mathematical Society, 1994.
2.E. Cela, The Quadratic Assignment Problem. Kluwer, 1998.
SITES:
1.Quadratic assignment Problem Library:

12. Multi-objective QAP (mQAP)
The multi-objective QAP (mQAP) with multiple flow matrices naturally models any facility layout problem that is concerned with the flow of more than one type of items or agents.
For example, in a hospital layout problem we may be concerned with simultaneously minimizing the flow/distance products of doctors on their rounds, of patients, of hospital visitors, and of pharmaceuticals and other equipment.

13. The mQAP proposed by Knowles and Corne [17] uses different flow matrices and keeps the same distance matrix.
Given n facilities and n locations, a nxn matrix A, where a(i,j) is the distance between locations i and j, and T number of nxn Bt matrices, t=1..T, where btr,s is the tth flow between facilities r and s, the mQAP can be stated as follows:

15. Representation !!!!

16. The 0-1 Multiple Knapsack Problem

19. Vehicle Routing Problem
The vehicle routing problem (VRP) is a combinatorial optimization and integer programming problem seeking to service a number of customers with a fleet of vehicles. Proposed by Dantzig and Ramser in 1959, VRP is an important problem in the fields of transportation, distribution, and logistics.

20. The Vehicle Routing Problem (VRP) is a generic name given to a whole class of problems in which a set of routes for a fleet of vehicles based at one or several depots must be determined for a number of geographically dispersed cities or customers. The objective of the VRP is to deliver a set of customers with known demands on minimum-cost vehicle routes originating and terminating at a depot. In the two figures below we can see a picture of a typical input for a VRP problem and one of its possible outputs:

21. VRP