Storage and Warehousing - UB

1

Storage and Warehousing

Chapter 10

Warehouse Functions

Provide temporary storage of

goods

Put together customer orders

Serve as a customer service

facility

Protect goods

Segregate hazardous or

contaminated materials

Perform value-added

services

Inventory

Elements of a Warehouse

Storage Media

Material Handling System

Building

Storage Media

Block Stacking

Stacking frames

Stool like frames

Portable (collapsible) frames

Cantilever Racks

Storage Media (Continued)

Selective Racks

Single-deep

Double-deep

Multiple-depth

Combination

Drive-in Racks

Drive-through Racks


Mobile Racks

Flow Racks

Push-Back Rack

2


Racks for AS/RS

Combination Racks

Modular drawers (high

density storage)

Racks for storage and

building support

Storage and Retrieval Systems

Person-to-item

Item-to-person

Manual S/RS

Semi-automated S/RS

Automated S/RS

Aisle-captive AS/RS

Aisle-to-aisle AS/RS

Storage and Retrieval Systems

(cont) Storage Carousels

Vertical

Horizontal

Miniload AS/RS

Robotic AS/RS

High-rise AS/RS

(two motors)

Phoenix Pharmaceuticals

German company

founded in 1994

Receives supplies from

19 plants across

Germany and

distributes to drugstores

$400 million annual

turnover

Phoenix Pharmaceuticals

30% market share

Fill orders in < 30

minutes

87,000 items

61% pharmaceutical,

39% cosmetic

Phoenix Pharmaceuticals (cont.)

150-10,000 picks per month

Three levels of automation

Manual picking via flow-racks

Semi-automated using dispensers

Full automation via robotic AS/RS

3

AVS/RS RFID

Warehouse Problems

Design

Operational or Planning

Warehouse Design

Location

How many?

Where?

Capacity

Overall Layout

R

C C C C C C

EXIT

Warehouse Design Warehouse Design

Layout and Location of

Docks

Pickup by retail

customers?

Combine or separate

shipping and receiving?

Layout of road/rail

network

Room available for

maneuvering trucks?

Similar trucks or a

variety of them?

Truc

k

Dock

Face

Sawthoot dock

Truck

Dock berths

Totally enclosed

Truck

Dock berths

Straight in, Straight out

Enclosure

Outside

building

wall

Flush dockDock

Face

Canopy

Open dockDock

Face

4

Warehouse Design (cont)

Number of Docks

Shipping and receiving

combined or

separated?

Average and peak

number of trucks or rail

cars?

Average and peak

number of items per

order?

Seasonal highs and

lows

Types of load

handled? Sizes?

Shapes? Cartons?

Cases? Pallets?

Protection from

weather elements

Model for Rack Design

x, y are # of columns, rows of rack spaces

a, b are aisle space multipliers in x, y

directions

Minimizex(a1) y(b1)

2

Subject to xyz n

x,y int eger

Model for Rack Design (Cont)

In the relaxed problem,

xyz=n

x=n/yz

The unconstrained objective is

n(a1) /yz y(b1)

2

Model for Rack Design (Cont)

Taking derivative with respect to y, setting

equation to zero and solving, we get

n(a1)

2y 2zb1

2 0

x n(b1)

z(a1)and y

n(a1)

z(b1)

Rack Design Example

Consider warehouse

shown in figure 10.29

Assume travel

originates at lower left

corner

Assume reasonable

values for the aisle

space multipliers a, b

Rack Design Example (Cont)

Example 1: Determine length and width of

the warehouse so as to accommodate 2000

square storage spaces of equal area in:

3 levels

4 levels

5 levels

5

Rack Design Example Solution

Reasonable values for a, b are 0.5, 0.2

For the 3-level case,

x 2000(0.21)

3(0.51) 24

y 2000(0.51)

3(0.21) 29


(Cont)

Previous solution gives a total storage of

24x29x3=2088

Due to rounding, we get 88 more spaces

If inadequate to cover the area required for

lounge, customer entrance/exit and other

areas, the aisle space multipliers a, b must be

increased appropriately and the x, y values

recalculated


(Cont)

For the 4 level and 5 level case, the building

dimensions are 25x20 units and 18x23 units,

respectively

Easy to calculate the average distance

traveled - simply substitute a, b, x and y

values in the objective function

For 3-level case, average one-way distance =

35.4 units

Warehouse Design Model

Model Assumptions

1. The available total storage space is known.

2. The expected time a product spends on the

shelves is known. This is referred to as the dwell

time throughout this paper.

3. The cost of handling each product in each flow

is known.

4. The dwell time and cost have a linear

relationship.

5. The annual product demand rates are known.

6. The storage policies and material handling

equipment are known and these affect the unit

handling and storage costs.

Model Notation

Parameters i : Number of products i = 1, 2, …, n.

j : Type of material flow; j=1,2,3,4

i : Annual demand rate of product i in unit loads

Ai : Order cost for product i

Pi: Price per unit load of product i

pi : Average percentage of time a unit load of product i spends in reserve area

if product is assigned to material flow 3

qij : 1 when product i is assigned to material flow j=1, 2 or 4;

di 1 when product i is assigned to flow j=3, where di is the ratio of the

size of the unit load in reserve area to that in forward area and

di is the

largest integer greater than or equal to di

6

Model Notation

a,b,c : Levels of space available in the vertical dimension in each functional area,

a - cross-docking, b - reserve, c – forward

r: Inventory carrying cost rate

Hij : Cost of handling a unit load of product i in material flow j

Cij : Cost of storing a unit load of product i in material flow j per year

iS : Space required for storing a unit load of product i

TS : Total available storage space

Qi : Order quantity for product i (in unit loads)

Ti: Dwell time (in years) per unit load of product i

CDCD ULLL , : Lower and upper storage space limit for cross-docking area

LLF,ULF : Lower and upper storage space limit for forward area

LLR,ULR : Lower and upper storage space limit for reserve area

Model Notation

a,b,c : Levels of space available in the vertical dimension in each functional area,

a - cross-docking, b - reserve, c – forward

r: Inventory carrying cost rate

Hij : Cost of handling a unit load of product i in material flow j

Cij : Cost of storing a unit load of product i in material flow j per year

iS : Space required for storing a unit load of product i

TS : Total available storage space

Qi : Order quantity for product i (in unit loads)

Ti: Dwell time (in years) per unit load of product i

CDCD ULLL , : Lower and upper storage space limit for cross-docking area

LLF,ULF : Lower and upper storage space limit for forward area

LLR,ULR : Lower and upper storage space limit for reserve area

Decision Variables

ijX = 1 if product i is assigned to flow type j ; 0 otherwise

,, : Proportion of available space assigned to each functional area, - cross-

docking, - reserve, - forward

Model

Model

Minimize

2 qijHijiX ijj1

4

i1

n

+

qijCijQiX ij /2 j1

4

i1

n

(1)

14

1

j

ijX i (2)

QiSiX i1 /2 i1

n

aTS (3)

QiSiX i2 /2 i1

n

piQiSiX i3 i1

n

bTS (4)

(1 pi)QiSiX i3 /2 i1

n

QiSiX i4 /2 i1

n

cTS (5)

Model

1 (6)

LLCD a TS ULCD (7)

LLR b TS ULR (8)

LLF c TS ULF (9)

,, 0 (10)

X ij 0or1

i, j (11)

Spreadsheet Based AS/RS Design

Tool

Spreadsheet Based AS/RS Design

Tool

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Block Stacking

Simple formula to determine a near-optimal

lane depth assuming

goods are allocated to storage spaces using

the random storage operating policy

instantaneous replenishment in pre-

determined lot sizes

replenishment done only when inventory

excluding safety stock has been fully depleted

lots are rotated on a FIFO basis

Block Stacking (Cont)

withdrawal of lots takes place at a constant

rate

empty lot is available for use immediately

Let Q, w and z denote lot size in pallet loads,

width of aisle (in pallet stacks) and stack

height in pallet loads, respectively


Kind’s (1975) formula for near-optimal lane

depth, d

d Qw

zw

2


E.g., if lot size is 60 pallets, pallets are stacked 3

pallets high and aisle width is 1.7 pallet stacks, then

Verify optimality by checking the utilization for all

possible lane depths (a finite number)

d 60(1.7)

3

1.7

2 5 pallets


Several issues omitted in Kind’s formula.

Some examples

What if pallets withdrawn not at a constant

rate but in batches of varying sizes?

What if lots are relocated to consolidate pallets

containing similar items?

Storage Policies

Random

In practice, not purely random

Dedicated

Requires more storage space than random,

but throughput rate is higher because no time

is lost in searching for items

Cube-per-order index (COI) policy

Class-based storage policy

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Storage Policies (Cont)

Shared storage policy

Class based and shared storage policies are

between the two “extreme” policies - random

and dedicated

Class based policy variations

if each item is a class, we have dedicated

policy

if all items in one class, we have random

policy

Design Model for Dedicated Policy

Warehouse has p I/O points

m items are stored in one of n storage spaces

or locations

Each location requires the same storage

space

Item i requires Si storage spaces


(Cont)

Ideally, we would like

However, if LHS < RHS, add a dummy

product (m+1) to take up remaining spaces

Sii1

m

n

n Sii1

m


(Cont)

So, assume that the above equality holds

But, if RHS < LHS, no feasible solution

Model Parameters

fik trips of item i through I/O point k

cost of moving a unit load of item i to/from I/O

point k is cik

distance of storage space j from I/O point k is

dkj


(Cont)

Model Variable

binary decision variable xij specifying whether

or not item i is assigned to storage space j


(Cont)

Minimize

c ik f ikdkjk1

p

Si

x ijj1

n

i1

m

Subject to x ijj1

n

Si i 1,2,...,m

9


(Cont)

x iji1

m

1 j 1,2,...,n

x ij 0 or1, i 1,2,...,m, j 1,2,...,n


(Cont)

Substituting wij

c ik f ikdkjk1

p

Si

, the obj fn. is

Minimize wijx ijj1

n

i1

m


(Cont)

Model is generalized QAP

Can be solved via transportation algorithm

No need for binary restrictions in the model


- Example WH Layout

1 2 3 4

5 6 7 8

9 10 11 12

13 14 15 16


- Example (Cont)

3 I/O points located in middle of south, west

and north walls

4 items


Example [fik(cik)]1 2 3 Si

1 150(5) 25(5) 88(5) 3

2 60(7) 200(3) 150(6) 5

3 96(4) 15(7) 85(9) 2

4 175(15) 135(8) 90(12) 6

10


Example Solution (dkj)

1 2 3 4 5 6 7 8 9 1

0

1

1

1

2

1

3

1

4

1

5

1

6

1 5 4 4 5 4 3 3 4 3 2 2 3 2 1 1 2

2 2 3 4 5 1 2 3 4 1 2 3 4 2 3 4 5

3 2 1 1 2 3 2 2 3 4 3 3 4 5 4 4 5


Example Solution (wij)

1 2 3 … 15 16

1 1627 1272 1313 ... 1003 1442

2 1020 876 996 ... 1284 1668

3 1830 1308 1361 ... 1932 2559

4 2908 2470 2650 ... 1878 2675


- Example Solution (Cont)

2 3 3 2

2 2 1 2

4 4 4 1

4 4 4 1

Design Model for COI Policy

Consider special case of dedicated storage

policy model

All items use I/O points in same proportion

Cost of moving a unit load of item i is

independent of I/O point

Define Pk as % trips through I/O point k

No need for the first subscript in fik as well as

cik


(Cont)

Minimize

c i f idkjk1

p

Si

x ijj1

n

i1

m

Subject to x ijj1

n

Si i 1,2,...,m


(Cont)

x iji1

m

1 j 1,2,...,n

x ij 0 or1, i 1,2,...,m, j 1,2,...,n

11


(Cont)

Substituting w j Pkdkjk1

p

, the obj fn. is

Minimizec i f i

Siw jx ij

j1

n

i1

m

Design Model for COI Policy -

Solution

COI model easier than Dedicated Model

Rearrange “cost”, “distance” terms (cifi/Si), wj

in non-increasing and non-decreasing order

Match

Item corresponding to 1st element in ordered

“cost” list with storage spaces corresponding

to 1st Si elements in ordered “distance” list


Solution Second item with storage spaces

corresponding to next Sl elements, and so on

…

COI policy calculates inverse of the “cost”

term and orders elements in non-decreasing

order, of their COI values, thereby producing

the same result as above


Solution

Arranging cost and distance vectors in non-

increasing and non-decreasing order and

taking their product provides a lower bound

on cost function

Above algorithm is optimal


Example

Consider dedicated policy example

Ignore cik and fik data

Assume

all 4 items use 3 I/O points in same proportion

pallets moved/time period are 100, 80, 120 and

90

cost to move unit load through unit distance is

$1.00

Determine optimal assignment of items to

storage spaces


Example Solution

12


Example Solution

Sort [cifi/Si] values in non-increasing order

[60, 33.33, 16, 15], corresponding to items 3,

1, 2 and 4

Optimal storage space assignment

Item 1 to Storage Spaces 2, 5, 7

Item 2 to Storage Spaces 1, 3, 9, 11, 14

Item 3 to Storage Spaces 6, 10

Item 4 to Storage Spaces 4, 8, 12, 13, 15, 16


Example Solution

2 1 2 4

1 3 1 4

2 3 2 4

4 2 4 4

Design Model for Random Policy

Items stored randomly in empty and available

storage spaces

Each empty space has an equal probability of

being selected

Storage or retrieval may not be purely

random, but we assume so for model

Design Model for Random Policy

(Cont)

Problem Definition

Determine storage space layout so total

expected travel distance between each of n

storage spaces and p I/O points is minimized

Sum of distances of each storage space from

each I/O point is

dkjk1

p

Design Model for Random Policy-

Solution

Arrange spaces in non-decreasing order of

the sum of above distances

Pick the n closest storage spaces

n depends upon inventory levels of all items

n is less than that required under dedicated

policy

Design Model for Random Policy -

Example

Determine storage space layout for 56

storage spaces in a 140x70 feet warehouse

Random storage policy

Minimize total distance traveled

Each storage space is a 10x10 feet square

I/O point located in middle of south wall

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Example (Cont)


Example Solution

Calculate distance of all potential storage

spaces to the I/O point

Arrange them in non-decreasing order


Example Solution (Cont)

Largest distance traveled is 70 feet

Sum total distance traveled (2800) by number

of storage spaces (56) to get average

distance traveled = 50 feet


Example Solution (Cont)

70 70

70 60 60 70

70 60 50 50 60 70

70 60 50 40 40 50 60 70

70 60 50 40 30 30 40 50 60 70

70 60 50 40 30 20 20 30 40 50 60 70

Travel Time Models

For random policy, average distance traveled

When number of storage spaces are large,

calculating average distance can be tedious

dkjj1

n

k1

p

n

Travel Time Models (Cont)

If storage spaces are small relative to total

area, approximate average distance traveled

assume spaces are continuous points on a

plane

use the integral

1

A(x y)dxdy

0

Y

0

X

14


We assume in previous integral that

warehouse is in 1st quadrant

only one I/O point (at origin and SW corner)

distance metric of interest is rectilinear

Previous integral can be easily modified if

two or more I/O points

distance metric is not rectilinear

no restrictions on location of warehouse


Suppose designer interested in shape that

minimizes travel time

Then, depending upon number and location

of I/O points, distance metric, warehouse

shape can range from diamond to circle to

trapezium !!!


Models minimizing construction costs and

travel distance

Consider following assumptions

Warehouse shape is fixed - rectangle

Warehouse area = A

Construction cost is function of warehouse

perimeter - r[2(a+b)]

r is unit (perimeter) distance construction cost

a and b are warehouse dimensions


One I/O point at origin and SW corner

coordinates are (p, q)

cost for each unit distance traveled = c

Model

2r(a b) c1

A( x y )dxdy

q

qb

p

pa


Optimal value of a and b, given that

I/O point must be on or outside exterior

walls, i.e., p$ 0

warehouse area must be A square units

a Ac 8r

2c 8r

and b A

2c 8r

c 8r


Single command cycle

Dual or multiple command cycles

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Warehouse Operations

Warehouse operational problems

Sequence in which orders to be picked

How frequently orders picked from high-rise

storage area?

Batch picking or pick when order comes in?

Limit on number of items picked?

If so, what is the limit?

Operator assignment to stacker cranes

Warehouse Operations (Cont)

How to balance picking operator’s workload?

Release items from stacker crane into

sorting stations in batches or as soon as

items are picked?

Order picking consumes over 50% of the

activities in warehouse

Warehouse Operations (Cont)

Not surprising that order picking is the single

largest expense in warehouse operations

Since construction and operation of AS/RS

are very high,managers interested in

maximizing throughput capacity

Order Picking Sequence

Two basic picking methods

Order picking

Zone picking

Consider this:

An AS/R machine has two independent

motors

Movement in horizontal and vertical

directions simultaneously

Order Picking Sequence (Cont)

Time to travel from (xi, yi) to (xj, yj)

maxxi x j

h,yi y j

v

Order Picking Sequence Model

Minimize dijwijj1, j1

n

i1

n

Subject to wiji1,i j

n

1 j 1,2,...,n

wijj1, j i

n

1 i 1,2,...,n

ui u j nwij n 1 2 i j n

wij 0 or 1 i, j 1,2,...,n

ui arbirary real numbers

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Algorithms

Construction

Improvement

Hybrid


Algorithms (Cont)

2-opt

3-opt

Branch-and-bound

Simulated Annealing

Convex Hull

Convex Hull Algorithm - Phase 1

Find xmax and ymax

Delete points inside polygon formed by

xmax, ymax and origin

For each region, construct convex path

between extreme points

Sort points in regions 1 and 2 in

ascending order of x-coordinate


(Cont)

Sort region 3 points in descending order

Starting with 1st extreme point, compute V

for three consecutive points i, i+1, i+2 V= (yi+1-yi)(xi+2-xi+1)+(xi-xi+1)(yi+2-yi+1).

Repeat until other extreme point is reached

If V# 0, no convex hull with i, i+1, i+2

Otherwise, convex hull possible


(Cont)

ymax

xmax

Region 1

Region 3

Region 2

0


(Cont)

Using some or all of the sorted points in

regions 1, 2, and 3, three at a time,

generate convex hull (sub-tour)

Points not in sub-tour are considered in

phases 2 and 3.

If xmax = ymax following explanation still good

17


Insert points that maybe included in sub-tour

without increasing cost

Such free insertion points lie on a

parallelogram with two adjacent points in the

sub-tour as its corner


Insert points not included in the sub-tour in

phases 1 and 2 using minimal insertion cost

criteria

greedy hull

steepest descent hull

If no points left for insertion in phase 2 or 3,

phase 1 sub-tour is optimal

Simulated Annealing Algorithm

Set S, z, r, Tin, T= Tin; Tfin= 0.1Tin

Randomly select points i and j in S and

exchange their positions

If new solution S' has z’< z, set S = S', and z

= z’

Otherwise, set S= S' with probability e-d/T

Simulated Annealing Algorithm

(Cont)

Repeat Step 1 until number of new solutions

= 16 times the number of neighbors

Set T= rT. If T > Tfin, go to Step 1

Otherwise return S, and STOP

TSP Software Routing Problem

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