指導老師：林燦煌 博士 報告者：梁士明 200 5/4/25

1. A literature survey on planning and control of warehousing systemsby JEROEN P. van den BERGPart II 指導老師：林燦煌 博士 報告者：梁士明 2005/4/25 1

2. Unit-load retrieval systems • Author:Goetschalckx, Ratliff[19] introduce duration of stay for individual load as alternative of COI(cube-per-order index 訂單體積指標 ，計算物品空間需求與暢銷性的關係) 2

3. Unit-load retrieval systems Hausman et al.[3] introduce the cumulative demand function G(i)=i^s and show that a class-based policy with relatively few classes yields mean travel times that are close to those obtained by dedicated policy • i denotes a fraction of the products which contains the products with highest COI • s is a suitably chosen parameter, and s=0.139 if 20% products generates 80% of all demand 3

4. Unit-load retrieval systems Graves et al.[2] observe furture travel time reductions when aloowing dual command cycles • Extended from Hausman et al.[3] • Analytic computations using a continuous rack and discrete computations using a rack with 30x10 locations • Determine the expected cycle time for combination of storage policies、sequencing strategies、queue length of S/R requests 4

5. Unit-load retrieval systems Schwarz et al. verify the analytic results in [2],[3] with simulation • Closest Open Location rule is applied to select a location under randomize storage policy • Mean travel times with COL rule are comparable to analytic results which baes on arbitrary location selection 5

6. Closest Open Location 靠近出口法則(Closest Open Location)：將剛到達的商品指派到離出入口最近的空儲位上。 Refer:http://www.materialflow.org.tw/abstract/book4/chap3.html 6

7. Chebyshev(柴比雪夫) travel • S/R machines can often move simultaneously along horizontal and vertical paths at speeds vx and vz. To reach a location (x,z) from (0,0) requires the Chebyshev measure travel time max(x/vx,z/vz). If rl is the rack length and rh the rack height Chebyshev travel require rl vx = rh vz • Rectangular building designs with I/O points at the eand of each aisle are often optimal for Chebyshev travel Refer : http://www.rh.edu/~ernesto/C_S2001/mams/notes/mams14.html 7

8. Unit-load retrieval systems Guenov & Raeside[20] in experiments, an optimum tour with respect to Chebyshev travel may be up to 3% above the optimum for travel time with acceleration/deceleration 8

9. Unit-load retrieval systems Hwang & Lee[21] provide a travel time measure that include acceleration/deceleration Chang et al.[22] consider various travel speeds 9

10. Order-picking systems Organ pipe arrangement • Aisles closest to the center should carry the highest COI 10

11. Control of warehousing operations • Batching of orders • Routing and sequencing • Dwell point positioning Focus on AS/RS 11

12. Batching of orders • To reduce mean travel time per order • Orders in batch may not exceed the storage capacity of vehicle • Large batches give rise to response times • Orders at the far end of WH delayed • Trade-off between efficiency and urgency 12

13. Batching of orders Two trade-offs • Static approach: select a block with most urgent orders and find a batching to minimize travel time • Dynamic approach: assign due date to orders and release orders immediately, then establish a schedule that satisfies these due date 13

14. Batching of orders For static approach • select a seed order for batch • Expand the batch with orders that have proximity to seed order • Capacity can not be exceeded • Distinctive factor is the measure for the proximity of orders/batches 14

15. Routing and sequencing • Unit-load retrieval operations • Order-picking operations • Carousel operations • Relocation of storage 15

16. Unit-load retrieval operations Hausman et al.[3] only consider single command cycles 16

17. Unit-load retrieval operations Graves et al.[2] study the effects of dual command cycles and observe travel time reductions of up to 30% 17

18. Order-picking operations Ratliff & Rosenthal[56] present dynamic programming algorithm that solves TSP • In a parallel aisle warehouse with crossover aisles at both ends of ech aisle • Computation time is linear in the number of stops • Problem remains tractable if there are 3 crossovers per aisle 18

19. Traveling salesman problem(TSP) • The salesman have to visit the cities in his territory exactly once and return to the start point • find the itinerary(行程) of minimum cost 19

20. Order-picking operations Petersen[57] evaluates the performance of 5 routing heuristics in comparison with the algorithm of Ratliff & Rosenthal[56] • Best heuristics are on average 10% over optimal for various wh shapes, locations of I/O station and pick list sizes 20

21. Order-picking operations Goetschalckx & Ratliff[58] give algorithm for order-picking in WH with non-negligible aisle width • Savings of up to 30% are possible by picking both sides of the aisle 21

22. Order-picking operations Goetschalckx & Ratliff[59] propose a dynamic programming algorithm that the travel time of the order-picker is measured with the rectilinear metric • Determine the optimal stop position of vehicle when performing multiple picks per stop is allowed 22

23. Order-picking operations Gudehus[1] describes band heuristic • Rack is devides into 2 horizontal bands • Vehicle visit the locations of lower band on increasing x-coordinate • Subsequentlt, visit upper band on decreasing x-coordinate 23

24. Order-picking operations Golden & Stewart[60] • TSP for which travel times are measured by Euclidean metric has an optimal solution • Nodes on the boundary of the convex hull are visited in the same sequence 24

25. Convex hull(凸包) • 求最小凸多邊形(convex polygon,沒有凹陷位)將平面上給定的所有點包含在裡面 Refer :http://www.geocities.com/kfzhouy/Hull.html 25

26. Convex hull(凸包) • Akl & Toussaint[61] present a fast algorithm for finding the convex hull 26

27. Order-picking operations Bozer et al.[64] present that use convex hull of the rack locations as an initial subtour • Locations in the interior of hull are inserted • For Chebyshev & rectilinear metric some locations can be inserted without increasing the travel time • also present an improved version of the band heuristic that blocks out a central portion of the rack 27

28. Order-picking operations Hwang & Song[65] present a heuristic that considers the convex hull for Chebyshev travel and rectilinear hull for rectilinear travel to ensure safety of pickers • Below a predetermined height Chebyshev travel is performed • Above this height , rectilinear travel is performed 28

29. Order-picking operations Daniels et al.[66] consider the situation where products are stored at multiple location and picked freely. It’s not acceptable because • Propagates aging of the inventory (not FIFO) • Increases storage space requirements (multiple incomplete pallets) 29

30. Carousel operations Bartholdi and Platzman[67] present a linear time algorithm • Sequencing picks in single order • Assume time needed by robot to move between bins within the same carrier is negligible compared to the time rotating carousel to next carrier • Reduce the problem of finding shortest Hamiltonian path on a circle 30

31. Hamiltonian path 由數學家 Euler 提出的：西洋棋的騎士能否走完一個空棋盤的六十四格，而且每格只走過一次。這條路徑，在圖論上稱為「Hamiltonian path」 ，而每個格子稱為「vertex」，每個格子能向外走出的步數稱為「該vertex的degree」。 • Refer:http://episte.math.ntu.edu.tw/java/jav_knight/ 31

32. Carousel operations Wen and Chang[68] present 3 heuristics • Sequencing picks in single order • Time to move between bins may not be neglected • Based upon the algorithm in Bartholdi and Platzman[67] 32

33. Carousel operations Ghosh and Wells[69], van den Berg[70] present optimal pick sequence • Multiple orders • Dynamic programming algorithm • Sequence of orders is fixed • Sequence of picks in orders is free 33

34. Carousel operations Bartholdi and Platzman[67] present a heuristic for the problem with extra constraint • Order sequence is free • Picks within same order must be performed consecutively • Extra constraint: each order is picked along its shortest spanning interval 34

35. Carousel operations Van den Berg[70] presents a polynomial time algorithm that solve the problem with extra constraint to optimality • At most 1.5 revolutions of the carousel above a lower bound for the problem without extra constraint • Reveal that the upper bound of one revolution presented by Bartholdi and Platzman[67] for their heuristic is incorrect 35

36. Relocation of storage Jaikumar and Solomon[71] address the problem of relocating pallets with a high expectancy of retrieval to locations closer I/O station during off-peak hours • Assume there is sufficient time (travel time is omitted) • Present a algorithm to minimize the number of relocations 36

37. Relocation of storage Muralidharan et al.[72] suggest randomized location assignment • Combines benefits of randomized storage (less storage space) and class-based storage (less travel time) • Respect to their turnover rate during idle periods 37

38. Dwell point positioning Dwell point : the position the S/R machine resides when system is idle • Minimize the travel time from the dwell point to position of 1st transaction • If 1st operation is advanced, all operations within the sequence are completed earlier 38

39. Dwell point positioning Graves et al.[2] select the point at the I/O station and Park[73] shows the optimality • If the probability of the 1st operation after idle period being a storage is at least 0.5 39

40. Dwell point positioning Egbelu[74] presents LP-model that • Minimize the expected travel time • Minimize the maximum travel time to the 1st transaction 40

41. Dwell point positioning Egbelu and Wu[75] use simulation to evaluate the performance of several strategies 41

42. Dwell point positioning Hwang and Lim[76] treats this problem as a Facility Location Problem • Computational complexity is equivalent to sorting a set of numbers 42

43. Dwell point positioning Peters et al.[77] presents an analytic model based on expressios found by Bozerand White[78] 43

44. Question & Discussion