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指導老師：林燦煌 博士 報告者：梁士明 200 5/4/25PowerPoint Presentation

指導老師：林燦煌 博士 報告者：梁士明 200 5/4/25

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### Question & Discussion systems

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

Unit-load retrieval systems 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

Unit-load retrieval systems 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

Unit-load retrieval systems 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

Unit-load retrieval systems 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

Closest Open Location systems

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

Chebyshev( systems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

Unit-load retrieval systems 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

Unit-load retrieval systems systems

Hwang & Lee[21] provide a travel time measure that include acceleration/deceleration

Chang et al.[22] consider various travel speeds

9

Order-picking systems systems

Organ pipe arrangement

- Aisles closest to the center should carry the highest COI

10

Control of warehousing operations systems

- Batching of orders
- Routing and sequencing
- Dwell point positioning
Focus on AS/RS

11

Batching of orders systems

- 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

Batching of orders systems

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

Batching of orders systems

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

Routing and sequencing systems

- Unit-load retrieval operations
- Order-picking operations
- Carousel operations
- Relocation of storage

15

Unit-load retrieval operations systems

Graves et al.[2] study the effects of dual command cycles and observe travel time reductions of up to 30%

17

Order-picking operations systems

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

Traveling salesman problem(TSP) systems

- 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

Order-picking operations systems

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

Order-picking operations systems

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

Order-picking operations systems

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

Order-picking operations systems

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

Order-picking operations systems

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

Convex hull( systems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

Order-picking operations systems

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

Order-picking operations systems

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

Order-picking operations systems

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

Carousel operations systems

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

Hamiltonian path systems

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

Carousel operations systems

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

Carousel operations systems

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

Carousel operations systems

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

Carousel operations systems

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

Relocation of storage systems

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

Relocation of storage systems

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

Dwell point positioning systems

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

Dwell point positioning systems

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

Dwell point positioning systems

Egbelu[74] presents LP-model that

- Minimize the expected travel time
- Minimize the maximum travel time to the 1st transaction

40

Dwell point positioning systems

Egbelu and Wu[75] use simulation to evaluate the performance of several strategies

41

Dwell point positioning systems

Hwang and Lim[76] treats this problem as a Facility Location Problem

- Computational complexity is equivalent to sorting a set of numbers

42

Dwell point positioning systems

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

43

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