Elements of a Warehouse

  Storage and Warehousing

   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

Chapter 10 Warehouse Functions

   Storage Media  Material Handling System  Building

  Storage Media

   Block Stacking  Stacking frames

   Stool like frames  Portable (collapsible) frames

   Inventory Elements of a Warehouse

  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  Push-Back Rack

   Cantilever Racks Storage Media (Continued)  Racks for AS/RS 

  Combination Racks

  Receives supplies from 19 plants across Germany and distributes to drugstores

  Semi-automated using dispensers

   Manual picking via flow-racks 

   Three levels of automation

  150-10,000 picks per month

  

  Phoenix Pharmaceuticals (cont.)

  39% cosmetic

   87,000 items  61% pharmaceutical,

  minutes

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

  turnover

   $400 million annual

  

   Modular drawers (high

  founded in 1994

  Phoenix Pharmaceuticals  German company

  High-rise AS/RS (two motors)

   Horizontal  Miniload AS/RS  Robotic AS/RS 

   Vertical

  Aisle-to-aisle AS/RS Storage and Retrieval Systems (cont)  Storage Carousels

   Manual S/RS  Semi-automated S/RS  Automated S/RS  Aisle-captive AS/RS 

  Item-to-person

   Person-to-item 

  Storage and Retrieval Systems

  Racks for storage and building support

  

  density storage)

   Full automation via robotic AS/RS

AVS/RS

  RFID Warehouse Problems Warehouse Design

   Location

   Design

   How many?

   Operational or Planning _R

   Where?  Capacity  Overall Layout

  C C C C C C _EXIT Warehouse Design Warehouse Design _{building Outside Enclosure}

   Layout and Location of _{wall Truck}

  Docks _Truck

   Pickup by retail Flush dock _{Dock Face Dock berths Dock berths} customers? _{Canopy Totally enclosed Straight in, Straight out Truc}

   Combine or separate _{k Dock} shipping and receiving? Face _{Dock Open dock Face}

   Layout of road/rail Sawthoot dock network

   Room available for maneuvering trucks?

   Similar trucks or a variety of them? Warehouse Design (cont) Model for Rack Design  Number of Docks  Seasonal highs and x(a

  1)  y(b 1) lows

  Minimize  Shipping and receiving  combined or Types of load

  2 handled? Sizes? separated?

  Subject to xyz  n

  Shapes? Cartons?  Average and peak

  Cases? Pallets? number of trucks or rail x, y int eger

   Protection from cars? weather elements

   Average and peak

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

  number of items per

  

  a, b are aisle space multipliers in x, y order? directions

  ฀

Model for Rack Design (Cont) Model for Rack Design (Cont)

   In the relaxed problem,  Taking derivative with respect to y, setting

  equation to zero and solving, we get xyz=n x=n/yz

  

  The unconstrained objective is n(a 1) 1 b

    0

  2 2y z

  2 n(a

  1)/ yz  y(b 1) 1) 1) n(b n(a

    x and y

  1) 1) z(a z(b

  2 ฀

  ฀ Rack Design Example Rack Design Example (Cont)  Consider warehouse

   Example 1: Determine length and width of shown in figure 10.29

  the warehouse so as to accommodate 2000  Assume travel square storage spaces of equal area in:

  originates at lower left

   3 levels

  corner

   4 levels  Assume reasonable

  values for the aisle  5 levels space multipliers a, b Rack Design Example Solution

   Reasonable values for a, b are 0.5, 0.2 

   1. The available total storage space is known. 

     1 when product i is assigned to flow j=3, where d _i is the ratio of the size of the unit load in reserve area to that in forward area and ฀ d _i

  ฀ d _i

  P _i : Price per unit load o f product i ฀ p _i : Average percentage of time a unit load of product i spends in reserve area if product is assigned to material flow 3 ฀ q _ij : 1 when product i is assigned to material flow j=1, 2 or 4;

  A _i : Order cost for product i ฀

   : Annual demand rate of product i in un it loads ฀

  1, 2, …, n. j : Type of material flow; j=1,2,3,4 _i

  Model Notation Parameters i : Number of products i =

  6. The storage policies and material handling equipment are known and these affect the unit handling and storage costs.

   5. The annual product demand rates are known. 

  4. The dwell time and cost have a linear relationship.

  

  3. The cost of handling each product in each flow 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.

  Model Assumptions

  For the 3-level case, ฀ x

   Due to rounding, we get 88 more spaces  If inadequate to cover the area required for

   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

  lounge, customer entrance/exit and other areas, the aisle space multipliers a, b must be increased appropriately and the x, y values recalculated

  For 3-level case, average one-way distance = 35.4 units Warehouse Design Model

  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

  

    is the largest intege r greater than or equa l to d _i

• – forward

   b TS (4) ฀ (1  p _i )Q _i S _i

  X i  (2) ฀

  Q _i S _i

  X _i1 /2   _{i 1} ⁿ

    a TS (3)

  ฀ Q _i S _i

  X _{i 2} /2   _{i 1} ⁿ

    p _i Q _i S _i

  X _{ i3  1 i} ⁿ 

  X _i3 /2   _{i 1} ⁿ 

  

   Q _i S _i

  X _i4 /2   _{i 1} ⁿ 

   c TS (5) Model

   1      (6) ฀ LL _{CD  a}  TS  UL _CD (7)

  ฀ LL _R

   b  TS  UL _R (8) ฀ LL _F

   c  TS  UL _F (9) ฀ ,,  0 (10) ฀ X _ij  0or1

  ฀ i, j (11) Spreadsheet Based AS/RS Design Tool

  1 ⁴ ₁   _{ j} ^ij

  ฀ ^{ij ij i ij}   _{j 1} ⁴  _{i 1} ⁿ

  Model Notation ฀ a,b,c : Levels of space available in the vertical dimension in each functional area, a - cross-docking, b - reserve, c

  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

   r : Inventory carrying cost rate ฀

  H _ij : Cost of handling a unit load of product i in material flow j ฀

  C _ij : Cost of storing a unit load of product i in material flow j per year _i S : Space required for storing a unit load of product i ฀

  TS : Total available storage space ฀

  Q _i : Order quantity for product i (in unit loads) ฀

  T _i : Dwell time (in years) per unit load of product i _{CD CD} UL LL , : Lower and upper storage space limit for cross-docking area ฀

  LL _F ,UL _F : Lower and upper storage space limit for forward area ฀

  LL _R ,UL _R : Lower and upper storage space limit for reserve area

  

฀

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

   _{i 1} ⁿ 

  

฀

C _ij : Cost of storing a unit load of product i in material flow j per year _i S : Space required for storing a unit load of product i

  

฀

TS : Total available storage space

  

฀

Q _i : Order quantity for product i (in unit loads)

  

฀

T _i : Dwell time (in years) per unit load of product i _{CD CD} UL LL , : Lower and upper storage space limit for cross-docking area

  

฀

LL _F

  ,UL _F : Lower and upper storage space limit for forward area

฀

LL _R

  ,UL _R : 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 ฀ ^{ij ij i ij j 1 4}

  Spreadsheet Based AS/RS Design Tool Block Stacking

   Simple formula to determine a near-optimal

  Storage Policies

  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?

  

  ฀ d

  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

   60(1.7) 3 

  possible lane depths (a finite number)

  lane depth assuming 

  rate

  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

   empty lot is available for use immediately

   Verify optimality by checking the utilization for all

   Let Q, w and z denote 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

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

  pallets high and aisle width is 1.7 pallet stacks, then

2 Block Stacking (Cont)

  Storage Policies (Cont)

  

   So, assume that the above equality holds  But, if RHS < LHS, no feasible solution  Model Parameters

   f ik

  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 c

  ik  distance of storage space j from I/O point k is d kj

  Design Model for Dedicated Policy (Cont)

  Model Variable  binary decision variable x ij

      

  specifying whether or not item i is assigned to storage space j

  Design Model for Dedicated Policy (Cont) ฀

  Minimize c ik f ik d kj k 1 p

   S i

       

        x ij j 1 n

   i 1 m

   Subject to x ij j 1 n

    Design Model for Dedicated Policy (Cont)

   S i i 1 m

   Shared storage policy 

  or locations

  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

   Each location requires the same storage

  ฀ n

  space

   Item i requires S

  i storage 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

    S i i 1,2,...,m

  

Design Model for Dedicated Policy Design Model for Dedicated Policy

(Cont) (Cont) p

    m c f d

    ik ik kj

   x

  1 j 1,2,...,n ij k 1

    

   Substituting w , the obj fn. is ij i 1

    S i

    x  0 or 1, i 1,2,...,m, j 1,2,...,n ij

    m n

  Minimize w x ij ij

    i 1 j 1

  ฀ ฀

  

Design Model for Dedicated Policy Design Model for Dedicated Policy

(Cont)

• Example WH Layout

   Model is generalized QAP  Can be solved via transportation algorithm  No need for binary restrictions in the model

  5

  6

  7

  8

  9

  10

  11

  12

  13

  14

  15

  16 Design Model for Dedicated Policy Design Model for Dedicated Policy

• Example (Cont) Example [f (c )]

  ik ik

  1

  2

  3 S

  i

  

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

  1 150(5) 25(5) 88(5)

  3

  and north walls

  

  4 items

  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 Design Model for Dedicated Policy

Example Solution (d ) Example Solution (w ) kj ij

  1 2 3 4 5 6 7 8 9 1

  1

  1

  1

  1

  1

  1

  1

  2

  3

  15

  16 …

  1

  2

  3

  4

  5

  6 1 1627 1272 1313 ... 1003 1442 1 5 4 4 5 4 3 3 4 3 2 2 3 2 1 1 2 2 1020 876 996 ... 1284 1668 2 2 3 4 5 1 2 3 4 1 2 3 4 2 3 4 5 3 1830 1308 1361 ... 1932 2559 3 2 1 1 2 3 2 2 3 4 3 3 4 5 4 4 5 4 2908 2470 2650 ... 1878 2675

  Design Model for Dedicated Policy Design Model for COI Policy

• Example Solution (Cont)

   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

  2

  2

  1

  2 

  Define P as % trips through I/O point k k

   No need for the first subscript in f as well as

  4

  4

  4

  1 ik c ik

  4

  4

  4

  1 Design Model for COI Policy Design Model for COI Policy (Cont)

  (Cont) p

    m c f d

    i i kj m n

   x 1 j 1,2,...,n ij

   k 1

    Minimize x i 1 ij

     

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

      n

  Subject to x  S i 1,2,...,m ij i

   j 1

  

฀

฀

  Design Model for COI Policy (Cont) ฀

   Above algorithm is optimal

  l

  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

  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

  Design Model for COI Policy - Example

  Design Model for COI Policy - Solution  Second item with storage spaces

  

  Consider dedicated policy example

   Ignore c

  ik and f ik 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

  corresponding to next S

  st S i elements in ordered “distance” list

  Substituting w j

   COI model easier than Dedicated Model 

   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

   Design Model for COI Policy - Solution

  Rearrange “cost”, “distance” terms (c i f i

  element in ordered “cost” list with storage spaces corresponding to 1

  /S i

  ), w j in non-increasing and non-decreasing order

   Match

  

  Item corresponding to 1

  st

  storage spaces Design Model for COI Policy Example Solution Design Model for COI Policy - Example Solution

  

   Design Model for Random Policy- Solution

  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 ฀ d kj k 1 p

  

  storage spaces

  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 empty space has an equal probability of

  4 Design Model for Random Policy  Items stored randomly in empty and available

  Sort [c i f i

  4

  /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

  Design Model for COI Policy Example Solution

  2

  1

  2

  1

  4

  3

  1

  4

  2

  3

  2

  4

  4

  2

   Each storage space is a 10x10 feet square  I/O point located in middle of south wall Design Model for Random Policy - Example (Cont) Design Model for Random Policy - Example Solution

   Calculate distance of all potential storage

   k

  1 A (x  y)dxdy Y 

   use the integral ฀

  plane

  If storage spaces are small relative to total area, approximate average distance traveled  assume spaces are continuous points on a

  

  Travel Time Models (Cont)

   n

  1 p

  1 n

  spaces to the I/O point

  calculating average distance can be tedious ฀ d kj j

   When number of storage spaces are large,

  For random policy, average distance traveled

  

  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

  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}

   Largest distance traveled is 70 feet  Sum total distance traveled (2800) by number

  Design Model for Random Policy - Example Solution (Cont)

   Arrange them in non-decreasing order

  X  Travel Time Models (Cont)

   We assume in previous integral that

  walls, i.e., p $ 0

  Model

฀

2r(a  b)  c

  1 A ( x  y )dxdy q q b

   p p a 

  Travel Time Models (Cont)

  

  Optimal value of a and b, given that  I/O point must be on or outside exterior

   warehouse area must be A square units ฀ a

   cost for each unit distance traveled = c

   A c  8r

  2c  8r   

     and b  A 2c  8r c  8r

     

    Travel Time Models (Cont)

   Single command cycle 

  

  coordinates are (p, q)

   warehouse is in 1st quadrant  only one I/O point (at origin and SW corner)  distance metric of interest is rectilinear

  of I/O points, distance metric, warehouse shape can range from diamond to circle to trapezium !!!

   Previous integral can be easily modified if

   two or more I/O points  distance metric is not rectilinear  no restrictions on location of warehouse

  Travel Time Models (Cont)

   Suppose designer interested in shape that

  minimizes travel time

   Then, depending upon number and location

  Travel Time Models (Cont)

   One I/O point at origin and SW corner _

   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 Travel Time Models (Cont)

  Dual or multiple command cycles Warehouse Operations

   Warehouse operational problems

  ฀

  , y

  j

  )

  ฀ max x i

   x j h

  , y i

   y j v

     

     

  Order Picking Sequence Model

  Minimize d _ij w _{ij j} 1, j1 ⁿ

  ) to (x

   _i 1 ⁿ

   Subject to w _{ij i}

  1,i j ⁿ 

  1 j 1,2,...,n

  w _{ij j 1, ji} ⁿ 

  1 i 1,2,...,n

  u _i

   u _j  nw _ij  n 1 2  i  j  n

  w _ij

   0 or 1 i, j 1,2,...,n

  j

  i

   Sequence in which orders to be picked  How frequently orders picked from high-rise

  largest expense in warehouse operations

  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

   Since construction and operation of AS/RS

  , y

  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 (x i

  u _i arbirary real numbers following explanation still good

  max

  Sort region 3 points in descending order

  = y

  max

  If x

   Points not in sub-tour are considered in phases 2 and 3. 

  regions 1, 2, and 3, three at a time, generate convex hull (sub-tour)

  Convex Hull Algorithm - Phase 1 (Cont)  Using some or all of the sorted points in

  Region 2

  Region 1 Region 3

   Otherwise, convex hull possible Convex Hull Algorithm - Phase 1 (Cont) y _max x _max

   Repeat until other extreme point is reached  If V # 0, no convex hull with i, i+1, i+2

  )(x _i+2 -x _i+1 )+(x _i -x _i+1 )(y _i+2 -y _i+1 ).

   V= (y _i+1

  for three consecutive points i, i+1, i+2

  st

  Convex Hull Algorithm - Phase 1 (Cont) 

  Order Picking Sequence Algorithms

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

  between extreme points 

   For each region, construct convex path

  , y max and origin

  max and y max x max

   Find x

  Convex Hull Algorithm - Phase 1

   Branch-and-bound  Simulated Annealing  Convex Hull

  3-opt

   2-opt 

  Order Picking Sequence Algorithms (Cont)

   Hybrid

  Improvement

   Construction 

• y _i

  Otherwise return S, and STOP TSP Software Routing Problem

  = 0.1T in

  

  fin , go to Step 1

   Set T= rT. If T > T

  = 16 times the number of neighbors

   Repeat Step 1 until number of new solutions

  Simulated Annealing Algorithm (Cont)

   Otherwise, set S= S' with probability e

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

  

  exchange their positions

   Randomly select points i and j in S and

  ; T fin

  Convex Hull Algorithm - Phase 2

  in , T= T in

   Set S, z, r, T

  phase 1 sub-tour is optimal Simulated Annealing Algorithm

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

   steepest descent hull

   greedy hull

  Insert points not included in the sub-tour in phases 1 and 2 using minimal insertion cost criteria

  

  parallelogram with two adjacent points in the sub-tour as its corner Convex Hull Algorithm - Phase 3

   Such free insertion points lie on a

  Insert points that maybe included in sub-tour without increasing cost

  

• d /T

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