Major Topics Already Covered

1
Major Topics Already Covered

• Major roles of Warehouses in contemporary supply
  chains
• Warehouse processes and the organization of the
  material flow
• Warehouse equipment
• Overview of the warehouse design and control
  problem(s)
• Warehouse Activity Profiling

2
Next Issues

• Storage configuration and storage policies
• the forward/reserve problem
• order-picking batching, zoning, and routing
• Pallet-building
• Warehouse layout
• Configuring and controlling automated storage and
  retrieval equipment
• Cross-docking

3
Warehouse Storage Configuration and Storage
Policies

• Bibliography
• Bartholdi Hackman Chapter 6
• Francis, McGinnis, White Chapter 5
• Askin and Standridge Sections 10.3 and 10.4

4
Storage Policies

• Main Issue Decide how to allocate the various
  storage locations of a uniform storage medium to
  a number of SKUs.

5
Types of Storage Policies

• Dedicated storage Every SKU i gets a number of
  storage locations, N_i, exclusively allocated to
  it. The number of storage locations allocated to
  it, N_i, reflects its maximum storage needs and
  it must be determined through inventory activity
  profiling.
• Randomized storage Each unit from any SKU can by
  stored in any available location
• Class-based storage SKUs are grouped into
  classes. Each class is assigned a dedicated
  storage area, but SKUs within a class are stored
  according to randomized storage logic.

6
Location Assignment under dedicated storage
policy

• Major Criterion driving the decision-making
  process Enhance the throughput of your storage
  and retrieval operations by reducing the travel
  time ltgt reducing the travel distance
• How? By allocating the most active units to the
  most convenient locations...

7
Convenient Locations

• Locations with the smallest distance d_j to the
  I/O point!
• In case that the material transfer is performed
  through a forklift truck (or a similar type of
  material handling equipment), a proper distance
  metric is the, so-called, rectilinear or
  Manhattan metric (or L1 norm) d_j
  x(j)-x(I/O) y(j)-y(I/O)
• For an AS/RS type of storage mode, where the S/R
  unit can move simultaneously in both axes, with
  uniform speed, the most appropriate distance
  metric is the, so-called Tchebychev metric (or L?
  norm)
  
• d_j max (x(j)-x(I/O),y(j)-y(I/O))

8
Active SKUs

• SKUs that cause a lot of traffic!
• In steady state, the appropriate activity
  measure for a given SKU i
• Average visits per storage location
• (number of units handled per unit of time) /
• (number of allocated storage locations)
• TH_i / N_i

9
Problem Representation
Location
SKU
N_1
1
1
1
c_ij (TH_i/N_i)d_j
N_i
i
1
j
N_S
S
L
1
10
Problem Formulation

• Decision variables x_ij 1 if location j is
  allocated to SKU I 0 otherwise.
• Formulation
• min S_i S_j (TH_i/N_i) d_j x_ij
• s.t.
• ? i, S_j x_ij N_i
• ? j, S_i x_ij 1
• ? i, j, x_ij ? 0,1 gt x_ij ? 0

11
Remarks

• The previous problem representation corresponds
  to a balanced transportation problem Implicitly
  it has been assumed that L S_i N_i
• For the problem to be feasible, in general, it
  must hold that
• L ? S_i N_i
• If L - S_i N_i gt 0, the previous balanced
  formulation is obtained by introducing a
  fictitious SKU 0, with
• N_0 L - S_i N_i and TH_0 0

12
A fast solution algorithm

• Rank all the available storage locations in
  increasing distance from the I/O point, d_j.
• Rank all SKUs in decreasing turns, TH_i/N_i.
• Move down the two lists, assigning to the next
  most highly ranked SKU i, the next N_i locations.

13
Example
A 20/102
B 15/5 3
C 10/2 5
D 20/5 4
A
A
A
A
A
B
B
A
D
D
D
A
B
A
B
A
C
C
D
D
A
B
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Locating the I/O point

• In many cases, this location is already
  predetermined by the building characteristics,
  its location/orientation with respect to the
  neighboring area/roads/railway tracks, etc.
• Also, in the case of an AS/RS, this location is
  specified by the AS/RS technical/operational
  characteristics.
• In case that the I/O point can be placed at will,
  the ultimate choice should seek to enhance its
  proximity to the storage locations.

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Locating the I/O point Example 1
Option A
16
Locating the I/O point Example 2
Option A
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Option C
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I
17
Example 2 (cont.)

• Option A U-shaped or cross-docking configuration
• amplifies the convenience/inconvenience of
  close/distant locations
• appropriate for product movement with strong ABC
  skew
• provides flexibility for interchanging between
  shipping and receiving docking capacity
• allows for dual command operation of forklifts,
  reducing, thus, the deadhead traveling
• minimizes truck apron and roadway
• Option C Flow-through configuration
• attenuates the convenience difference among
  storage locations
• conservative design more reasonably convenient
  storage locations but fewer very convenient
• more appropriate for extremely high volume
• preferable when the building is long and narrow
• limits the opportunity for efficiencies for dual
  command operations

18
Storage Sizing

• Dedicated storage
• How many storage locations, N_i, should be
  dedicated to each SKU i?
• Randomized Storage
• How many storage locations, N, should be employed
  for the storage of the entire SKU set?
• Class-based storage
• How should SKUs be organized into classes?
• How many storage locations, N_k, should be
  dedicated to each SKU class k?

19
Possible Approaches to Storage Sizing under
Dedicated Storage

• Issue resolved/predetermined through operational
  (e.g., replenishment) policies, contractual
  agreements, etc.
• Service-level type of analysis
• Determine the number of storage locations, N_i to
  be assigned to each SKU i in a way that tends to
  control the probability that there will be a lack
  of storage space in any operational period (e.g.,
  day).
• Cost-based Analysis
• Select N_is in a way that minimizes the total
  operational cost over a given horizon, taking
  into consideration the cost of owning and
  operating the storage space and equipment, and
  also any additional costs resulting from space
  shortage and/or the need to contract additional
  storage space.

20
Sizing dedicated storage based on service level
requirements

• F_i(Q_i) Probstor. loc. requested by SKU i in
  a single period ? Q_i
• i.e., the cumulative distribution function of
  the storage locations requested by SKU i during
  any single operational period (day)
• Assuming independence of daily storage
  requirements across SKUs, for an assignment of
  N_i locations to each SKU i,
• Probno storage shortages in a single day
  ?_i F_i(N_i)
• and
• Prob1 or more storage shortages 1 - ?_i
  F_i(N_i)

21
Sizing dedicated storage based on service level
requirements (cont.)

• Formulation I Fixed service level, P
• min ?_i N_i
• s.t.
• ?_i F_i(N_i) ? P
• N_i ? 0 ? i
• Formulation II Fixed storage space, S
• max ?_i F_i(N_i)
• s.t.
• ?_i N_I ? S
• N_i ? 0 ? i

22
Sizing randomized storage based on service
level requirements

• F(Q) Probstor. loc. requested by all SKUs in
  a single period ? Q
• i.e., the cumulative distribution function of
  the storage locations requested by all SKUs
  during any single operational period (day)
• For a storage size of N locations
• ProbNo storage shortage in a single period
  F(N)
• Problem Formulations
• Formulation I Fixed service level, P
• min N
• s.t.
• F(N) ? P
• N ? 0
• Formulation II Fixed storage space, S
• max F(N)
• s.t.
• 0 ? N ? S

23
Class-Based Storage Sizing and Location
Assignment

• Divide SKUs into classes, using ABC (Pareto)
  analysis, based on their number of turns
  TH_i/N_i.
• Determine the required number of storage
  locations for each class C_k
• ad-hoc adjustment of the total storage
  requirement of the class SKUs
• N_k p ?_i?C_k N_i
• Class-based service-level type of analysis,
  e.g.,
• Assign to each class the requested storage
  locations, prioritizing them according to their
  number of turns,
• TH_k ( ?_i?C_k TH_i)/N_k

min ?_k N_k s.t. ?_k F_k(N_k) ? P N_k ?
0 ? k
24
A simple cost-based model for (dedicated)
storage sizing

• Model-defining logic Assuming that you know your
  storage needs d_ti, for each SKU i, over a
  planning horizon T, determine the optimal storage
  locations N_i for each SKU i, by establishing a
  trade-off between the
• fixed and variable costs for developing this set
  of locations, and operating them over the
  planning horizon T, and
• the costs resulting from any experienced storage
  shortage.

25
A simple cost-based model for (dedicated)
storage sizing (cont.)

• Model Parameters
• T length of planning horizon in time periods
• d_ti storage space required for SKU i during
  period t
• C_0 discounted present worth cost per unit
  storage capacity owned during the
  planning horizon T
• C_1t discounted present worth cost per unit
  stored in owned space during period
  t
• C_2t discounted present worth cost per unit
  stored in leased space or per unit
  of space shortage during period t
• Model Decision Variables
• N_i owned storage capacity for SKU i
• Model Objective
• min TC (N_1,N_2,,N_n)
• S_i C_0 N_i S_t C_1t min(d_ti, N_i) C_2t
  max(d_ti - N_i, 0)

26
A fast solution algorithm for the case of
time-invariant costs

• For each SKU i
• Sequence the storage demands appearing in the
  d_ti, t1,T, sequence in decreasing order.
• Determine the frequency of the various values in
  the ordered sequence obtained in Step 1.
• Sum the demand frequencies over the sequence.
• When the obtained partial sum is first equal to
  or greater than
• C C_0/ (C_2-C_1)
• stop the optimum capacity for SKU i, N_i,
  equals the corresponding demand level.

27
Example

• Problem Data
• N1 T6 d lt 2, 3, 2, 3, 3, 4,gt C_0 10,
  C_1 3, C_2 5
• Solution

C C_0/(C_2-C-1) 10/(5-3) 5
gt N 2
28
A probabilistic version

• Additional data
• p(d_ti) probability mass function for the
  storage requirement of SKU i during period t
• New objective function
• min E TC (N_1,N_2,,N_n)
• S_i C_0 N_i S_t C_1 Emin(d_ti, N_i) C_2
  Emax(d_ti - N_i, 0)
• S_i C_0 N_i
• S_t C_1 S_d_ti ? N_i d_ti p(d_ti) S_d_ti gt
  N_i N_i p(d_ti)
• C_2 S_d_ti gt N_i (d_ti - N_i) p(d_ti)
  

29
A probabilistic version (cont.)

• Model Properties
• Separable
• Piece-wise linear
• convex
• Optimality Condition
• N_i Q_ij optimal ltgt
• S_t Fc_ti(Q_ij) ? C_0 / (C_2 - C_1) ? S_t
  Fc_ti(Q_i,j1) ltgt
• S_t 1-F_ti (Q_ij) p(Q_ij) ? C_0 / (C_2 -
  C_1) ?
• S_t 1-F_ti (Q_i,j1)p(Q_i,j1)
• where F_ti( ) is the cdf for the storage
  requirements of SKU i in period t.

30
Storage Configuration and Policiesfor Unit
Load warehouses Topics covered

• Storage Policies Assigning storage locations of
  a uniform storage medium to the various SKUs
  stored in that medium
• Dedicated
• Randomized
• Class-based
• Criterion Maximize productivity by reducing the
  traveling effort / cost
• The placement of the I/O point(s)
• Criterion Maximize productivity by reducing the
  traveling effort / cost

31
Storage Configuration and Policiesfor Unit
Load warehouses Topics covered (cont.)

• Storage sizing for various SKUs Determine the
  number of storage locations to be assigned to
  each SKU / group of SKUs.
• Criterion
• provide a certain (or a maximal) service level
• minimize the total (spaceequipmentlaborshortage
  ) cost over a planning horizon
• Next major theme Storage Configuration for
  better space exploitation
• floor versus rack-based storage for
  pallet-handling warehouses
• determining the lane depth (mainly for randomized
  storage)
• (based on Bartholdi Hackman, Section 6.3)

32
Determining the Employment (and Configuration) of
Rack-based storage

• Basic Logic
• For each SKU,
• compute how many pallet locations would be
  created by moving it into rack of a given
  configuration
• compute the value of the created pallet
  locations
• move the sku into rack if the value it creates is
  sufficient to justify the rack.
• Remark In general, space utilization will be
  only one of the factors affecting the final
  decision on whether to move an SKU into rack or
  not. Other important factors can be
• the protection that the rack might provide for
  the pallets of the considered SKU
• the ability to support certain operational
  schemes, e.g., FIFO retrieval
• etc.

33
Examples on evaluating the efficiencies from
moving to rack-based storage

• Case I Utilizing 3-high pallet rack for an SKU
  of N4 (pallets), which is not stackable at all.
• Current footprint 4 pallet positions
• Introducing a 3-high rack in the same area
  creates 3x412 position, 8 of which are available
  to store other SKUs. What are the gains of
  exploiting these new locations vs the cost of
  purchasing and installing the rack?
• Case II Utilizing a 3-high pallet rack for an
  SKU with N30 (pallets), which are currently
  floor-stacked 3-high, to come within 4 ft from
  the ceiling.
• Current footprint 10 pallet positions
• Introducing a 3-high rack does not create any new
  positions, and it will actually require more
  space in order to accommodate the rack structure
  (cross-beams and the space above the pallets,
  required for pallet handling)

34
Determining an efficient lane depth(in case of
randomized storage)

• A conceptual characterization of the problem
• More shallow lanes imply more of them, and
  therefore, more space is lost in aisles (the size
  of which is typically determined by the
  maneuvering requirements of the warehouse
  vehicles)
• On the other hand, assuming that a lane can be
  occupied only by loads of the same SKU, a deeper
  lane will have many of its locations utilized
  over a smaller fraction of time (honeycombing).
• So, we need to compute an optimal lane depth,
  that balances out the two opposite effects
  identified above, and minimizes the average floor
  space required for storing all SKUs.

Aisle
35
Notation

• w pallet width
• d pallet depth
• g gap between adjacent lanes
• a aisle width
• x lane depth
• n number of SKUs
• N_i max storage demand by SKU i
• z_i column height for SKU I
• lane footprint (gw)(dxa/2)

36
Key results

• Assuming that the same lane depth is employed
  across all n SKUs, under floor storage, the
  average space consumed per pallet is minimized by
  a lane depth computed approximately through the
  following formula
• x_opt ?(a/2dn)?_i (N_i /z_i)
• The optimal lane depth for any single SKU i,
  which is stackable z_i pallets high, is
• x_opt ?(a/2d)(N_i /z_i)
• Assuming that the same lane depth is employed
  across all n SKUs, under rack storage, the
  average space consumed per pallet is minimized by
  a lane depth computed approximately through the
  following formula
• x_opt ?(a/2dn)?_i N_i