Facility Design-Week12 Warehouse Operation

Facility Design-Week12

Warehouse OperationAnastasia L. Maukar

1

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

2

Elements of a Warehouse• Storage Media• Material Handling System• Building

3

Storage Media• Block Stacking • Stacking frames• Stool like frames• Portable (collapsible) frames

• Cantilever Racks

4

Storage Media (Continued)• Selective Racks• Single-deep• Double-deep• Multiple-depth• Combination

• Drive-in Racks• Drive-through Racks

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Storage Media (Continued)• Mobile Racks• Flow Racks• Push-Back Rack

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Storage Media (Continued)• Racks for AS/RS• Combination Racks• Modular drawers (high

density storage)• Racks for storage and

building support

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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

8

Storage and Retrieval Systems (cont)

• Storage Carousels• Vertical• Horizontal

• Miniload AS/RS• Robotic AS/RS• High-rise AS/RS (two

motors)

9

Phoenix Pharmaceuticals

• German company founded in 1994• Receives supplies from 19

plants across Germany and distributes to drugstores• $400 million annual

turnover

10

Phoenix Pharmaceuticals• 30% market share• Fill orders in < 30

minutes• 87,000 items• 61% pharmaceutical,

39% cosmetic

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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

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AVS/RS

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RFID

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Warehouse Problems• Design• Operational or Planning

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Warehouse Design

• Location• How many?• Where?• Capacity

• Overall Layout

R

C C C C C C

EXIT

16

Warehouse Design

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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?

Truck

DockFace

Sawthoot dock

Truck

Dock berthsTotally enclosed

Truck

Dock berthsStraight in, Straight out

Enclosure

Outsidebuilding

wall

Flush dockDockFace

Canopy

Open dockDockFace

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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

19

Model for Rack Design

• x, y are # of columns, rows of rack spaces• a, b are aisle space multipliers in x, y directions

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Minimize x(a 1) y(b 1)2

Subject to xyz nx, y int eger

Model for Rack Design (Cont)• In the relaxed problem,

xyz=nx=n/yz

• The unconstrained objective is

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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

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n(a 1)2y 2z

b 1

20

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

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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

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Rack Design Example Solution• Reasonable values for a, b are 0.5, 0.2• For the 3-level case,

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x 2000(0.2 1)

3(0.5 1)24

y 2000(0.5 1)3(0.2 1)

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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, b must be increased appropriately and the x, y values recalculated

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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, x and y values in the objective function• For 3-level case, average one-way distance = 35.4 units

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Warehouse Design Model

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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.

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Model Notation

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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 areaif 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 thesize of the unit load in reserve area to that in forward area and

di is thelargest integer greater than or equal to di

Model Notation

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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

32

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

H ij : 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

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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)QiSi X i3 /2 i1

n

QiSiX i4 /2 i1

n

cTS (5)

Model

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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

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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

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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

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Block Stacking (Cont)• Kind’s (1975) formula for near-optimal lane depth, d

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d Qw

z

w2

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)

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d 60(1.7)

3

1.72

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?

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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

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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

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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

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Sii1

m

n

n Sii1

m

Design Model for Dedicated Policy (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

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Design Model for Dedicated Policy (Cont)• Model Variable• binary decision variable xij specifying whether or not item i is

assigned to storage space j

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Design Model for Dedicated Policy (Cont)

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Minimizec ik f ikdkj

k1

p

Si

x ijj1

n

i1

m

Subject to x ijj1

n

Si i 1,2,...,m


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x iji1

m

1 j 1,2,...,n

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


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Substituting wij c ik f ikdkj

k1

p

Si

, the obj fn. is

Minimize wij x ijj1

n

i1

m

Design Model for Dedicated Policy (Cont)• Model is generalized QAP• Can be solved via transportation algorithm• No need for binary restrictions in the model

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Design Model for Dedicated Policy - Example WH Layout

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1 2 3 4

5 6 7 8

9 10 11 12

13 14 15 16

Design Model for Dedicated Policy - Example (Cont)• 3 I/O points located in middle of south, west and north walls• 4 items

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Design Model for Dedicated Policy Example [fik(cik)]

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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

Design Model for Dedicated Policy Example Solution (dkj)

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1 2 3 4 5 6 7 8 9 10111213141516

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

Design Model for Dedicated Policy Example Solution (wij)

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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

Design Model for Dedicated Policy - Example Solution (Cont)

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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

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Design Model for COI Policy (Cont)

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Minimizec i f idkj

k1

p

Si

x ijj1

n

i1

m

Subject to x ijj1

n

Si i 1,2,...,m


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x iji1

m

1 j 1,2,...,n

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


61

Substituting w j Pkdkjk1

p

, the obj fn. is

Minimize c i f i

Si

w j x ijj1

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

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• 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

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• 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

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Design Model for COI Policy - 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

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Design Model for COI Policy Example Solution

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Design Model for COI Policy - 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

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Design Model for COI Policy Example Solution

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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

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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

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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

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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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Design Model for Random Policy - Example (Cont)

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Design Model for Random Policy - Example Solution• Calculate distance of all potential storage spaces to the I/O

point• Arrange them in non-decreasing order

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Design Model for Random Policy - 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

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Design Model for Random Policy - Example Solution (Cont)

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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

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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

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1A

(x y)dxdy0

Y

0

X

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

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Travel Time Models (Cont)• 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 !!!

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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)]• r is unit (perimeter) distance construction cost• a and b are warehouse dimensions

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Travel Time Models (Cont)• One I/O point at origin and SW corner • coordinates are (p, q)

• cost for each unit distance traveled = c• Model

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2r(a b) c 1A

( x y )dxdyq

qb

p

pa

Travel Time Models (Cont)• 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

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a A c 8r2c 8r

and b A 2c 8r

c 8r

Travel Time Models (Cont)• 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

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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

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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

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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

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Order Picking Sequence (Cont)• Time to travel from (xi, yi) to (xj, yj)

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maxx i x j

h,

y i y j

v

Order Picking Sequence Model

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Minimize dijw ijj1, j1

n

i1

n

Subject to wiji1,ij

n

1 j 1,2,...,n

wijj1, ji

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

Order Picking Sequence Algorithms• Construction• Improvement• Hybrid

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Order Picking Sequence Algorithms (Cont)• 2-opt• 3-opt• Branch-and-bound• Simulated Annealing• Convex Hull

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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

93

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

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95

ymax

xmax

Region 1

Region 3

Region 2

0


• 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

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Convex Hull Algorithm - Phase 2

• 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

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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

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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

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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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TSP Software

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Routing Problem

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Multimedia CD for Distribution Center Design

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Facility Design-Week12 Warehouse Operation

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