1. Optimization of order-picking using a revised minimum spanning table method
盧坤勇
國立聯合大學電子工程系

2. Minimum spanning tree : MST
Problem statement:
Given a connected graph
G = (V, E) ,
where V={v0, v1, …, vn-1} is the set of vertices and E V × V is the set of edges.
MST is a connected sub-graph of G of minimum cost with no cycles.

3. Traditional MST solution
Linear programming method
Integer programming
S.T:

4. Some well-known heuristic algorithms
Kruskal, 1956
Prim, 1959
Sollin, 1965

5. Revised MST algorithm(cont.)
Step 1: listing the cost relationships of vertices by two dimensional matrices( n × n matrices)
Step 2: choosing the minimum cost for each row and marking the minimum one from choose cost (e.g. Cij)
Step 3: connecting the vertices of xi and xj and deleting the ith row and jth column from the matrices
Step 4: repeating step 2 and 3, until deleting all rows and columns, or all vertices are selected

6. Revised MST algorithm(cont.)
Step 5: detecting and marking the results by isolated node, tree, and cycle
5-1 : if single tree only exists, stop
5-2 : if a cycle tree exists, then de-cycling in a tree with minimum cost
Step 6: connecting all isolated nodes and trees by some heuristic rules: e.g. Branch and Bound, GA, etc.

7. Example
Step 1 X1 X2 X3 X4 X5 X6 X7 * 9 8 4 5 6 3 2 1 7

8. Example
Step 2
Minimum cost / row X1 X2 X3 X4 X5 X6 X7 * 9 8 4 5 6 3 2 1 7

9. Example (cont.) Step 3 X1 X2 X3 X4 X5 X6 X7 * 9 8 4 5 6 3 2 11 7 1X1 X2 X3 X4 X5 X6 X7 * 9 8 4 5 6 3 2 11 7 1
x4→x5 

10. Example (cont.) Step 4 X1 X2 X3 X4 X5 X6 X7 * 9 8 4 5 6 3 2 11 7 1 12X1 X2 X3 X4 X5 X6 X7 * 9 8 4 5 6 3 2 11 7 1
x4→x5  12
x5→x6  

11. Example (cont.) Step 4 X1 X2 X3 X4 X5 X6 X7 * 9 8 4 5 46 6 3 23 2 11 7X1 X2 X3 X4 X5 X6 X7 * 9 8 4 5 46
x1→x7   6
Isolated vertex  3 23 2
x3→x4   11 7 1
x4→x5  12
x5→x6   24
x6→x3 35
x7→x1  

12. Example (cont.) Vertex index x1 x2 x3 x4 x5 x6 x7 Sequence 6 7 3 1 2 4
Step 5
Vertex index x1 x2 x3 x4 x5 x6 x7
Sequence 6 7 3 1 2 4 5
Destination
Cycle index
Results:
Two cycles: x4-x5-x6-x3, x7-x1
One isolated vertex

13. Example (cont.) Step 6 6-1 de-cycling x4 → x5 →x6 →x3 →

14. Example (cont.) Step 6 6-2 connecting
1. choosing isolated vertex firstly x4 x3 x1 x7 x2 6 4 9 8
Alternative solutions: x2→x3, x3→x2, x7→x2

15. Example (cont.) Step 6 2. Connecting results x2 → x3 → x4 → x5 → x6

16. Example (cont.) x4 x1 x2 6 * x3 3 Step 6 3. Connecting other trees
{x1,x7,x2} , {x4,x5,x6,x3} x4 x1 x2 6 * x3 3
Minimum cost
Alternative solutions: x3→x1

17. Example (cont.) Step 6 4. Connecting results
x4 → x5 → x6 → x3 →x1 → x7 → x2
Total cost : 15
5. Optimizing the results by using Branch and Bound or GA algorithms

18. Application of GA to optimize the generalized results
Procedure 1 ：Initialize the population
Procedure 2 ：Evaluate the fitness
Procedure 3 ：Parents selection
Procedure 4 ：Genetic operation

19. Application of GA to optimize the generalized results(cont.)
Procedure 1 ：Initialize the population
Step1. Collect the groups against the results of MST and give a sequence number, ex : G1={2},
G2={1,7},
G3={4,5,6,3}
Step2. Initialize parameters : index q=1, a population size s and population P = {Ø }.
Step3. Randomly produce a integer number Pq to represent the group , ex : a number 1 represents the group G1.

20. Application of GA to optimize the generalized results (cont.)
Procedure 1 ：Initialize the population
Step4. If Pq is feasible, go to step 5, or else go to step 3.
Step5. If Pq is different from any previous individuals, then P = P + {Pq} , q=q+1, or else go to step 3.
Step6. If q > s, then P = {p1, p2, …, Ps} is the initial population and stop; or else go to step3.

21. Application of GA to optimize the generalized results (cont.)
Procedure 2 ： Evaluate the fitness
Step 1. Initialize a constant c, decrement rate d and evaluation value E.
Step 2. Order the chromosomes in the decreasing order of evaluation value.
Step 3. Based on E, calculate the fitness value Fi, which starts at c, ane reduces linearly with decrement rate r, Fi = c+ (i-1) r, i = (1,2,…,s) where s is the size of the population.

22. Application of GA to optimize the generalized results (cont.)
Procedure 3 ： Parents selection
Step 1. Compute the fitness value of all the population members, Fsum = ,s is the population size.
Step 2. Initialize, index i = 0 and a counter F = 0.
Step 3. Randomly generate a real number f [0, Fsum].
Step 4. i = i + 1, F = F + fi .

23. Application of GA to optimize the generalized results (cont.)
Procedure 3 ： Parents selection
Step 5. If F > f, then return selected position i and stop; or else go to step 4.
Step6 . Select the first chromosome if n is smaller than or equal to the sum of cumulative probability of proceeding chromosomes.

24. Application of GA to optimize the generalized results (cont.)
Procedure 4 ： Genetic operation
Step 1. Generate a bit string.
Step 2. Check those numbers of parent1 against the ordered list of the bit string.
Step 3. If those numbers against digit 1 from parent1, move those numbers from parent1 to offspring at the same position..

25. Application of GA to optimize the generalized results (cont.)
Procedure 4 ： Genetic operation
Step 4. Check those numbers against digit 0 from parent1 and then find those numbers occurring on parent2.
Step5 . Move those numbers to unfilled positions of the offspring in the same sequence of parent2.

26. Application of GA to optimize the generalized results (cont.)
Example : crossover operation
Bit string
Parent
Offspring
Parent

27. Application of GA to optimize the generalized results (cont.)
Example : mutate operation
Parent | |
Offspring | |