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 S.T:

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 * 9 8 4 5 6 3 2 1 7

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

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

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

Example (cont.) Step 4 X1 X2 X3 X4 X5 X6 X7 * 9 8 4 5 46 6 3 23 2 11 7   X1 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  

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

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

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

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

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

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

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

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.

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.

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.

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 .

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.

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..

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.

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

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