optimization of order-picking using a revised minimum spanning table method

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

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.

Traditional MST solution

Linear programming method

Integer programming

ijji

ij xCMin

S.T:

Gjijix

Ejixx

Ejix

Gjxx

ij

jiij

ij

jijiji

ij

,,,1,0

),(,1

),(,1

,

Some well-known heuristic algorithms

Kruskal, 1956

Prim, 1959

Sollin, 1965

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

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.

Example

Step 1

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

Example

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

Step 2 Minimum cost / row

X1 X2 X3 X4 X5 X6 X7 X1 * 9 8 4 5 9 4 4 X2 9 * 4 6 4 9 9 4 X3 3 4 * 2 3 5 4 2 X4 4 8 5 * 11 7 8 1 x4→x5

X5 5 9 8 8 * 1 4 1 X6 3 4 2 7 3 * 6 2 X7 3 4 8 5 9 4 * 3

Step 3

Example (cont.)

X1 X2 X3 X4 X5 X6 X7 X1 * 9 8 4 5 9 4 4 X2 9 * 4 6 4 9 9 4 X3 3 4 * 2 3 5 4 2 X4 4 8 5 * 11 7 8 1 x4→x5

X5 5 9 8 8 * 12 4 1 x5→x6 X6 3 4 2 7 3 * 6 2

X7 3 4 8 5 9 4 * 3

Step 4

Example (cont.)

X1 X2 X3 X4 X5 X6 X7 X1 * 9 8 4 5 9 46 4 x1→x7

X2 9 * 4 6 4 9 9 4Isolated vertex

X3 3 4 * 23 3 5 4 2 x3→x4 X4 4 8 5 * 11 7 8 1 x4→x5

X5 5 9 8 8 * 12 4 1 x5→x6 X6 3 4 24 7 3 * 6 2 x6→x3

X7 35 4 8 5 9 4 * 3 x7→x1

Step 4

Example (cont.)

Step 5

Example (cont.)

Vertex index

x1 x2 x3 x4 x5 x6 x7

Sequence 6 7 3 1 2 4 5

Destination x7 x2 x4 x5 x6 x3 x1

Cycle index 2 3 1 1 1 1 2

Results: Two cycles: x4-x5-x6-x3, x7-x1 One isolated vertex

Step 6

Example (cont.)

6-1 de-cyclingx4 → x5 →x6 →x3 →

x4 → x5 →x6 →x3

x1 x7 x7→x1

Step 6

Example (cont.)

6-2 connecting1. choosing isolated vertex firstly

x4 x3 x1 x7

x2 6 4 9 9

8 4 9 4

x2

Alternative solutions: x2→x3, x3→x2, x7→x2

Step 6

Example (cont.)

2. Connecting results

x2 → x3 → x4 → x5 → x6

x4 → x5 → x6 → x3 → x2

x1 → x7 → x2

Step 6

Example (cont.)

3. Connecting other trees {x1,x7,x2} , {x4,x5,x6,x3}

Alternative solutions: x3→x1

x4 x1

x2 6 *

x3 * 3

Minimum cost

Step 6

Example (cont.)

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

Procedure 1 ： Initialize the population

Procedure 2 ： Evaluate the fitness

Procedure 3 ： Parents selection

Procedure 4 ： Genetic operation

Application of GA to optimize the generalized results


Application of GA to optimize the generalized results(cont.)

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.


Application of GA to optimize the generalized results (cont.)

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.

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.



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 .

s

iif

1



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.



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..



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.

Example : crossover operation


Bit string 1 0 0 1 0 1 0 0 1 1

Parent1 7 4 8 1 3 6 9 10 2 5

Offspring 7 3 8 1 10 6 4 9 2 5

Parent2 3 5 2 8 7 10 4 1 6 9

Example : mutate operation


Parent1 5 4 8 | 1 7 2 3 | 10 2 5

Offspring 5 4 8 | 3 7 2 1 | 10 2 5

optimization of order-picking using a revised minimum spanning table method

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