nkfust 1 a literature survey on planning and control of warehousing systems by jeroen p. van den...

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NKFUST

A literature survey on planning and control

of warehousing systems

by JEROEN P. van den BERG

Part II

指導老師：林燦煌博士報告者：梁士明

2005/4/25

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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 )

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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

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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

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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

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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

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• 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

Chebyshev( 柴比雪夫 ) travel

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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

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

Chang et al.[22] consider various travel speeds

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Order-picking systems

Organ pipe arrangement• Aisles closest to the center should

carry the highest COI

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Control of warehousing operations

• Batching of orders• Routing and sequencing• Dwell point positioning

Focus on AS/RS

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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

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Batching of ordersTwo 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

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Batching of ordersFor static approach1. select a seed order for batch2. 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

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Routing and sequencing

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

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Unit-load retrieval operations

Hausman et al.[3] only consider single command cycles

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Unit-load retrieval operations

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

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Order-picking operationsRatliff & 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

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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

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Order-picking operationsPetersen[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

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Order-picking operationsGoetschalckx & 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

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Order-picking operationsGoetschalckx & 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

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Order-picking operationsGudehus[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

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Order-picking operationsGolden & 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

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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

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Convex hull( 凸包 )• Akl & Toussaint[61] present a fast

algorithm for finding the convex hull

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Order-picking operationsBozer 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

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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

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Order-picking operationsDaniels 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)

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Carousel operationsBartholdi 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

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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/

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Carousel operationsWen 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]

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Carousel operationsGhosh 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

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Carousel operationsBartholdi 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

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Carousel operationsVan 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

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Relocation of storageJaikumar 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

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Relocation of storageMuralidharan 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

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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

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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

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Egbelu[74] presents LP-model that• Minimize the expected travel time• Minimize the maximum travel time to

the 1st transaction

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

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

• Computational complexity is equivalent to sorting a set of numbers

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

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nkfust 1 a literature survey on planning and control of warehousing systems by jeroen p. van den...

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