A literature survey on planning and control of warehousing systems
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指導老師:林燦煌 博士 報告者:梁士明 200 5/4/25 - PowerPoint PPT Presentation

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A literature survey on planning and control of warehousing systems by JEROEN P. van den BERG P art II. 指導老師:林燦煌 博士 報告者:梁士明 200 5/4/25. Unit-load retrieval systems.

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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




Unit load retrieval systems
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)


Unit load retrieval systems1
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


Unit load retrieval systems2
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


Unit load retrieval systems3
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


Closest open location
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。



Chebyshev travel
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


Unit load retrieval systems4
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


Unit load retrieval systems5
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


Order picking systems
Order-picking systems systems

Organ pipe arrangement

  • Aisles closest to the center should carry the highest COI


Control of warehousing operations
Control of warehousing operations systems

  • Batching of orders

  • Routing and sequencing

  • Dwell point positioning

    Focus on AS/RS


Batching of orders
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


Batching of orders1
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


Batching of orders2
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


Routing and sequencing
Routing and sequencing systems

  • Unit-load retrieval operations

  • Order-picking operations

  • Carousel operations

  • Relocation of storage


Unit load retrieval operations
Unit-load retrieval operations systems

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


Unit load retrieval operations1
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%


Order picking operations
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


Traveling salesman problem tsp
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


Order picking operations1
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


Order picking operations2
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


Order picking operations3
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


Order picking operations4
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


Order picking operations5
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


Convex hull
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


Convex hull1
Convex hull( systems凸包)

  • Akl & Toussaint[61] present a fast algorithm for finding the convex hull


Order picking operations6
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


Order picking operations7
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


Order picking operations8
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)


Carousel operations
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


Hamiltonian path
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/


Carousel operations1
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]


Carousel operations2
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


Carousel operations3
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


Carousel operations4
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


Relocation of storage
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


Relocation of storage1
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


Dwell point positioning
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


Dwell point positioning1
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


Dwell point positioning2
Dwell point positioning systems

Egbelu[74] presents LP-model that

  • Minimize the expected travel time

  • Minimize the maximum travel time to the 1st transaction


Dwell point positioning3
Dwell point positioning systems

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


Dwell point positioning4
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


Dwell point positioning5
Dwell point positioning systems

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