Elements of a Warehouse

(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

Storage Media (Continued)

Mobile Racks

Flow Racks

(2)

Storage Media (Continued)

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

(3)

AVS/RS

RFID

Warehouse Problems

Design

Operational or Planning

Warehouse Design

Location

How many?

Where?

Capacity

Overall Layout

R

C CC C C C EXIT

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 dock Dock Face Canopy

(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, yare # of columns, rows of rack spaces

a, bare aisle space multipliers in x, y

directions

฀

Minimize

x

(

a



1)



y

(

b



1)

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

(

a



1) /

yz



y

(

b



1)

2

Model for Rack Design (Cont)

Taking derivative with respect to y, setting equation to zero and solving, we get

฀

n

(

a



1)

2

y

2

z



b



1

2



0

x



n

(

b



1)

z

(

a



1)

and

y



n

(

a



1)

z

(

b



1)

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)

Rack Design Example Solution

Reasonable values for a, bare 0.5, 0.2

For the 3-level case,

฀

x



2000(0.2



1)

3(0.5



1)



24

y



2000(0.5



1)

3(0.2



1)



29

Rack Design Example Solution

(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, bmust be increased appropriately and the x, yvalues recalculated

Rack Design Example Solution

(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, xand 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 un it loads

฀

Ai: Order cost for product i

฀

Pi: Price per unit load o f 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

(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 iin material flowj

฀

Cij: Cost of storing a unit load of product iin material flowjper year

i

S: 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

CD CDUL

LL , : 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 iin material flowj

฀

Cij: Cost of storing a unit load of product iin material flowjper year

i

S: 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

CD CDUL

LL , : 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

ij

X = 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

M odel

Minimize

฀

2 qijHijiXij j1 4  i1 n  + ฀

qijCijQiXij/2

  j1 4  i1 n  (1) 1 4 1    j ij

X i (2)

฀

QiSiXi1/2

 

i1

n

 aTS (3)

฀

QiSiXi2/2

 

i1

n

  piQiSiXi3

i1

n

 bTS (4)

฀

(1pi)QiSiXi3/2

 

i1

n

  QiSiXi4/2

i1

n

 cTS (5)

Model

1      (6) ฀

LLCDaTSULCD (7)

฀

LLRbTSULR (8)

฀

LLFcTSULF (9)

฀

,, 0 (10)

฀

Xij0or1

฀

i,j (11)

Spreadsheet Based AS/RS Design

Tool

(7)

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, wand zdenote lot size in pallet loads, width of aisle (in pallet stacks) and stack height in pallet loads, respectively

Block Stacking (Cont)

Kind’s (1975) formula for near-optimal lane depth, d

฀

d



Qw

z



w

2

Block Stacking (Cont)

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

Block Stacking (Cont)

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

(8)

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 pI/O points

mitems are stored in one of nstorage spaces or locations

Each location requires the same storage space

Item irequires Sistorage spaces

Design Model for Dedicated Policy

(Cont)

Ideally, we would like

However, if LHS < RHS, add a dummy product (m+1) to take up remaining spaces

฀

S

_i

i1 m





n

฀

n Si

i1 m



 

 

 

Design Model for Dedicated Policy

(Cont)

So, assume that the above equality holds

But, if RHS < LHS, no feasible solution

Model Parameters

f_iktrips of item ithrough I/O point k

cost of moving a unit load of item ito/from I/O point kis c_ik

distance of storage space jfrom I/O point kis

d_kj

Design Model for Dedicated Policy

(Cont)

Model Variable

binary decision variable x_ijspecifying whether or not item iis assigned to storage space j

(Cont)

Minimize

c

_ik

f

_ik

d

_kj

k1

p



S

i













x

ij j1

n



i1

m

(9)

(Cont)

฀

x

_ij i1

m





1

j



1,2,...,

n

x

ij



0

or

1,

i



1,2,...,

m

,

j



1,2,...,

n

(Cont)

฀

Substituting w

_ij



c

_ik

f

_ik

d

_kj

k1 p



S

_i













,

the obj fn.

is

Minimize

w

ij

x

ij

j1 n



i1

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 [f

_ik

(c

_ik

)]

1 2 3 Si

(10)

Example Solution (d

_kj

)

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 (w

_ij

)

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

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 iis independent of I/O point

Define Pkas % trips through I/O point k

No need for the first subscript in fikas well as

c_ik

Design Model for COI Policy

(Cont)

Minimize

c

_i

f

_i

d

_kj

k1

p



S

i













x

ij j1

n



i1

m



(Cont)

x

ij i1

m





1

j



1,2,...,

n

(11)

(Cont)

฀

Substituting w

j



P

k

d

kj

k1

p



,

the obj fn

.

is

Minimize

c

i

f

i

S

i

w

j

x

ij j1

n



i1 m



-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_S

i elements in ordered “distance” list

Design Model for COI Policy

-Solution

Second item with storage spaces

corresponding to next S_lelements, 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

Design Model for COI Policy

-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 c_ikand f_ikdata

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

(12)

Design Model for COI Policy

-Example Solution

Sort [c_if_i/S_i] 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 pI/O points is minimized

Sum of distances of each storage space from each I/O point is

฀

d

_kj

k1

p



Design Model for Random

Policy-Solution

Arrange spaces in non-decreasing order of the sum of above distances

Pick the nclosest storage spaces

ndepends upon inventory levels of all items

nis 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

(13)

-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

Design Model for Random Policy

-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

฀

dkj j1

n



k1

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)

Travel Time Models (Cont)

 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 !!!

Travel Time Models (Cont)

 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)]

ris unit (perimeter) distance construction cost

aand bare warehouse dimensions

One I/O point at origin and SW corner

coordinates are (p, q)

cost for each unit distance traveled = c

 Model

฀

2

r

(

a



b

)



c

1

A

(

x



y

)

dxdy

q

qb



p

pa



Travel Time Models (Cont)

 Optimal value of aand b, given that

I/O point must be on or outside exterior walls, i.e., p$0

warehouse area must be Asquare units

a



A

c



8

r

2

c



8

r













and

b



A

2

c



8

r

c



8

r













 Single commandcycle

(15)

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)

฀

max

x

i



x

j

h

,

y

i



y

j

v













Order Picking Sequence Model

฀

Minimize dijwij j1,j1

n



i1

n



Subject to wij i1,ij

n



1 j1,2,...,n

wij j1,ji

n



1 i1,2,...,n uiujnwijn1 2ijn

wij0or1 i,j1,2,...,n

(16)

Order Picking Sequence

Algorithms

 Construction

 Improvement

 Hybrid

Order Picking Sequence

Algorithms (Cont)

 2-opt

 3-opt

 Branch-and-bound

 Simulated Annealing

 Convex Hull

Convex Hull Algorithm - Phase 1

 Find x_maxand y_max

 Delete points inside polygon formed by

xmax, ymaxand origin

 For each region, construct convex path between extreme points

Sort points in regions 1 and 2 in ascending order of x-coordinate

Convex Hull Algorithm - Phase 1

(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 2}

Convex Hull Algorithm - Phase 1

(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.

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 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

Convex Hull Algorithm - Phase 3

 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 iand jin Sand 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

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