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

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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

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Minimum spanning tree : MST

Problem statement:

Given a connected graph G = (V, E) ,

where V={v₀, v₁, …, v_n-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.



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Traditional MST solution

Linear programming method Integer programming

ij j

i

ij x

C Min





S.T:

G j

i j i x

E j

i x

x

E j

i x

G j

x x

ij

ji ij

ij

ji j i j

i

ij





























 

, , ,

1 , 0

) , ( , 1

,

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Some well-known heuristic algorithms

Kruskal, 1956 Prim, 1959

Sollin, 1965

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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 x_i and x_j and deleting the

i_th row and j_th column from the matrices

Step 4: repeating step 2 and 3, until deleting all rows and columns, or all vertices are selected

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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.

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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 *

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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

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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.)

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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.)

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X1 X2 X3 X4 X5 X6 X7

X1 * 9 8 4 5 9 4⁶ 4 ^x1→x7

X2 9 * 4 6 4 9 9 4 ^Isolatedvertex

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.)

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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

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Step 6

Example (cont.)

6-1 de-cycling

x4 → x5 →x6 →x3 →

x4 → x5 →x6 →x3

x1 x7 x7→x1

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Step 6

Example (cont.)

6-2 connecting

1. 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

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Step 6

Example (cont.)

2. Connecting results

x2 → x3 → x4 → x5 → x6 x4 → x5 → x6 → x3 → x2 x1 → x7 → x2

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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

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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

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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

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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 P_q to represent the group , ex : a number 1

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

Step4. If P_qis feasible, go to step 5, or else go to step 3.

Step5. If P_q is different from any previous

individuals, then P = P + {P_q} , q=q+1, or else go to step 3.

Step6. If q > s, then P = {p₁, p₂, …, P_s} is the initial population and stop; or else go to step3.

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Procedure 2 ： Evaluate the fitness

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

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 F_i, which starts at c, ane reduces linearly with

decrement rate r, F_i = c+ (i-1) r, i = (1,2,…,s) where s is the size of the population.

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Procedure 3 ： Parents selection

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

Step 1. Compute the fitness value of all the population members, F_sum = ,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, F_sum].

Step 4. i = i + 1, F = F + f_i.

 s i

fi 1



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Procedure 3 ： Parents selection

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

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.

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

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..

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

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.

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Example : crossover operation

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

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

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Example : mutate operation

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

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