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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={v 0, v 1, …, 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 S.T:

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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 X1X2X3X4X5X6X7 X1* X29*46499 X334*2354 X4485*178 X55988*14 X634273*6 X *

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Example X1X2X3X4X5X6X7 X1* X29* X334*23542 X4485*1781 X55988*141 X634273*62 X *3 Step 2 Minimum cost / row

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X1X2X3X4X5X6X7 X1* X29* X334*23542 X4485*1 781 x4→x5 X55988*141 X634273*62 X *3 Step 3 Example (cont.)

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X1X2X3X4X5X6X7 X1* X29* X334*23542 X4485*1 781 x4→x5 X55988* x5→x6 X634273*62 X *3 Step 4 Example (cont.)

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X1X2X3X4X5X6X7 X1* x1→x7 X29* Isolated vertex X334* x3→x4 X4485*1 781 x4→x5 X55988* x5→x6 X *62 x6→x3 X *3 x7→x1 Step 4 Example (cont.)

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Step 5 Example (cont.) Vertex index x1x2x3x4x5x6x7 Sequence Destinationx7x2x4x5x6x3x1 Cycle index 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 x7x7→x1

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Step 6 Example (cont.) 6-2 connecting 1. choosing isolated vertex firstly x4x3x1x7 x 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 x4x1 x26* x3*3 Minimum cost

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Step 6 Example (cont.) 4. Connecting results x4 → x5 → x6 → x3 →x1 → x7 → x2 Total cost : 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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Procedure 1 ： Initialize the population 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 represents the group G1.

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Procedure 1 ： Initialize the population Application of GA to optimize the generalized results (cont. ) Step4. If P q is 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 = {p 1, p 2, …, 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.

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

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Procedure 4 ： Genetic operation Application of GA to optimize the generalized results (cont. ) Step 4. Check those numbers against digit 0 from parent 1 and then find those numbers occurring on parent 2. Step5. Move those numbers to unfilled positions of the offspring in the same sequence of parent 2.

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

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

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