Design of the fast-pick area

1
Design of the fast-pick area

• Based on Bartholdi Hackman, Chpt. 8

2
The fast-pick or forward-pick or
primary-pick area
Primary picking
Restocking
Shipping
Receiving
Forward pick Area
Reserves picking
Reserves Area

3
The major trade-offs behind the establishment of
a forward pick area

• A forward pick area increases the pick density by
  concentrating a large number of SKUs within a
  small physical space.
• On the other hand, it introduces the activity of
  restocking.
• Also, in general, a forward pick area concerns
  the picking of smaller quantities and involves
  more sophisticated equipment than the picking
  activity taking place in the reserves area. So,
  its deployment requires some capital investment
  in equipment and (extra) space.

4
Major issues to be resolved

• Which SKUs to store in the fast-pick area? (2)
• How much of each SKU to store? (1)
• How large should be the fast-pick area? (3)

5
Available approaches

• Fluid models (Bartholdi Hackman, Chpt. 8)
• essentially volume-based analysis of the
  underlying activity
• more appropriate for SKUs handled in smaller
  packages (cases, cartons, etc.)
• allows the use of continuous quantities in the
  modeling and analysis
• Slotting (Bartholdi Hackman, Chpt. 9)
• Allows the explicit modeling of
• the geometry of the storage area
• the geometry of the stored packages
• any additional geometrical or logical constraints
• Much more computationally expensive

6
A fluid model for determining the optimal
allocation of forward-pick storage to a
pre-determined set of SKUs

• Given
• V Volume of entire forward-pick storage area
  (e.g., in cubic ft)
• f_i Flow of SKU i, (e.g., in cubic ft / year)
• c_r cost of each restock trip (/trip)
• Determine
• u_i storage volume to be allocated to SKU i,
  i1,,n (cubic ft)
• s.t. total restocking cost (rate) is minimized
  (/year)
• Additional assumptions
• Replenishment for each SKU occur at lots equal to
  u_i, and occur instantaneously upon the complete
  depletion of the previous lot
• The entire replenishment lot is transferred in
  one trip

7
Problem formulation

• min ?_i c_r (average number of trips per year
  for SKU i)
• ?_i c_r (f_i / u_i)
• s. t.
• ?_i u_i ? V
• u_i ?? 0, ? i
• Optimal Solution
• ? i, u_i (? f_i / ?_k ? f_k) V

8
Remarks

• The fraction of fast-pick storage devoted to SKU
  i
• ? f_i / ?_k ? f_k
• Optimal number of replenishment trips per year
  for SKU i
• f_i / u_i (? f_i ?_k ? f_k) / V
• Each unit of the fast-pick storage should be
  restocked at the same rate
• Optimal number of restocks per year per cubic ft
  for SKU i (f_i / u_i) / u_i f_i / (u_i)2
  (?_k ? f_k)2 / V2
• i.e., independent of i. This result can be used
  for a quick assessment of the optimality of the
  current allocation in a fast-pick area, by
  considering how (spatially) balanced is the
  replenishment effort.

9
Other heuristics used in practice for resolving
the fast-pick storage allocation problem

• Equal-Space Allocation Assign each SKU the same
  amount of space, i.e.,
• u_i V / n, ? i
• Equal-Time Allocation Assign each SKU an
  equal-time supply, so that each SKU incurs the
  same number of restocking trips per year.
• u_i (f_i / ?_k f_k) V, ? i
• Hence, number of trips per year for SKU i,
• f_i / u_i (?_k f_k) / V

10
Comparing the performance of the heuristics and
the optimal optimal solution

• Performance of the optimal solution
• ?_i f_i / u_i (?_i ? f_i)2 / V
• Performance of the equal-space allocation
  heuristic
• ?_i f_i / u_i n (?_i f_i) / V
• Performance of the equal-time allocation
  heuristic
• ?_i f_i / u_i n (?_i f_i) / V

11
A statistical assessment of the sub-optimality of
the equal-space/time allocation

• Perf. of heuristics / Perf. of optimal sol.
• ?_i f_i / n

(?_i ? f_i / n)2
Assume that each ? f_i is an independent sample
from a random variable Y with mean m and variance
s2. Then, the above ratio is approximated
by m2 s 2

1 CV2
m2
Hence, the more diverse the rates of flow of the
various SKUs, the more sub-optimal is the
performance of the two heuristics.
12
Accommodating minimum (and maximum) allocation
constraints

• (e.g., we cannot allocate to an SKU a volume less
  than that required for storing at least one unit)
• Solution algorithm for accommodating minimum
  allocation constraints
• Identify those SKUs that received less than
  their minimum required space, when solving the
  problem without considering these constraints.
• Increase the allocations of these deficient SKUs
  to their minimum requirements, and remove them as
  well as their allocated space from any further
  consideration.
• Re-allocate the remaining space among the
  remaining SKUs.
• A similar type of logic can be applied for the
  accommodation of constraints imposing a maximum
  allocation

13
Selecting the SKUs to be accommodated in the
fast-pick area

• To resolve this issue, one must quantify the net
  benefit of having the SKU in the fast-pick area
  vs. doing all the picking from the reserve.
• This is done as follows Let
• V Volume of entire forward-pick storage area
  (e.g., in cubic ft)
• f_i Flow of SKU i, (e.g., in cubic ft / year)
• c_r cost of each restock trip (/trip)
• s the saving realized when a pick is done from
  the forward area rather than the reserve
  (/pick)
• p_i the expected annual picks for SKU i
  (picks/year)
• u_i storage volume to be allocated to SKU i,
  i1,,n (cubic ft)
• Then, the net annual benefit of allocating
  fast-pick storage u_i to SKU i, is
• c_i(u_i)

0 if u_i 0
(/year)
sp_i - c_r(f_i / u_i) if u_i gt 0
14
Plotting the net benefit function
c_i(u_i)
(c_rf_i) / (sp_i) minimum volume to be
stored, if any
u_i
15
Problem Formulation

• max ?_i c_i(u_i)
• s.t.
• ?_i u_i ? V
• u_i ?? 0, ? i
• A near-optimality condition
• The SKUs that have the strongest claim to the
  fast-pick area are those with the greatest
  viscocities, p_i / ? f_i.

16
Algorithm for computing a near-optimal solution

• Sort all SKUs from most viscous to least (p_i /
  ? f_i)
• For k 0 to n (total number of SKUs)
• Compute the optimal allocation of the fast-pick
  storage if it accommodates only the first k SKUs
  of the ordering obtained in Step 1.
• Evaluate the corresponding total net benefit.
• Pick the value of k that provides the largest
  total net benefit.

17
Proving the near-optimality of the SKU selection
algorithm

• Theorem Choosing SKUs based on their viscocity
  p_i / ? f_i,
• will lead to an objective value z such that
• z - z ? net benefit of a single SKU ? max_i
  (sp_i)
• where z denotes the optimal objective value.
• When there are many SKUs, the net benefit
  associated with a single SKU will be a very
  small/negligible fraction of the overall net
  benefit.

18
Extending the basic models

• Storage by family The originally developed
  results apply directly, with p_i, f_i and u_i
  pertaining to each family rather than single SKU.
• Introducing a re-order point rop_i for each SKU i

• Necessary modifications
• annual number of restock trips for SKU i f_i /
  (u_i - rop_i)
• u_i ? rop_i
• Optimal allocation of volume V
• u_i rop_i (? f_i / ?_k ? f_k) (V - ?_i
  rop_i) , ? i

19
Extending the basic models (cont.)

• Setting a priori limits on the total number of
  picks from the forward area, or the total number
  of restock trips Simply reject solutions that
  violate these constraints during the execution of
  the SKU selection algorithm.
• Accounting for on-hand inventory levels If an
  SKU is selected to be included to the fast-pick
  area, store all of it in the fast pick area if
  the maximum volume on-hand for it is no greater
  than
• 2 c_rf_i / (sp_i)
• (Remember that according to opt. solution of the
  majorized optimization problem, an SKU that
  enters its linear section has its volume
  increased until it exits it, or some boundary
  condition occurs.)

20
Extending the basic models (cont.)

• Dealing with set-up costs
• c_i(u_i)

-m_i if u_i 0
s p_i - (c_r f_i / u_i) - M_i if u_i gt 0

• where
• m_i a disruption cost of moving SKU i out of
  the fast-pick area
• M_i a disruption cost of introducing SKU i in
  the fast-pick area

Use the same algorithms as before, but with the
updated formula for the computation of the annual
benefit
21
Determining the Optimal Size of the Fast-Pick Area

• Basic trade-off A larger fast-pick area means
  more SKUs in it at larger volumes, and
  therefore, more picks from it and less
  restocking, but at the same time, the cost per
  pick increases.
• An analytical formulation of the underlying
  optimization problem
• s g(V) where g( ) is a decreasing function of
  V
• Linear storage models
• s a - bV
• constitute a very good approximation of the
  dependency of savings per pick on the volume of
  the fast-pick area for fast-pick areas organized
  in a linear fashion, e.g., an aisle of flow rack.

22
Characterizing the optimal storage size for
linear models of storage

• Theorem For linear models of storage (e.g.,
  adding bays to an aisle of flow rack), the
  optimum size of the fast pick area is given by
• V ? c_r ?_(i1)k ? f_i / ? ( b ?_(i1)k
  p_i)
• for some number k of the most viscous skus and
  where b is the decreasing rate of the pick
  savings per volume unit of the fast pick-area.

23
An algorithm for optimizing volume size, SKU
set, and space allocation of a fast-pick area for
small-item picking

• Rank all the n candidate SKUs in decreasing
  viscocity
• For k0 to n, consider the set of the k most
  viscous SKUs and compute
• the optimal storage size Vk, corresponding to
  this SKU selection
• (e.g., Vk V ? c_r ?_(i1)k ? f_i / ? ( b
  ?_(i1)k p_i)
• the optimal allocation of Vk to the
  corresponding SKU sub-set (e.g., for i1 to k,
  u_ik (? f_i / ?_j1k ? f_j) Vk )
• the resulting total benefit
• (e.g., ?_i1k s(Vk) p_i - c_r f_i / u_ik
  )
• Pick the value of k, denoted by k, that
  corresponds to the maximal total benefit set V
  V(k) and
• u_i u_i(k), for i1 to k 0, otherwise

24
Designing a fast-pick area for pallet storage

• Key observation When material is stored in
  pallets in the fast-pick area, each replenishment
  trip will correspond to a single unit load
• gt In this case, a more accurate measure for the
  resulting replenishment trips is the number of
  pallets moved through the fast-pick area (instead
  of f_i/u_i that we used for small-item picking).

25
A typical operating policy for case-pick-from-pal
let operation

• If no pallets are in the fast-pick area, then all
  picks are from the bulk storage.
• If some but not all pallets are in the fast-pick
  area, then all picks for less-than-pallet
  quantities are from the fast-pick area, and all
  picks for full-pallet quantities are from the
  bulk storage area.
• If all the pallets are in the fast-pick area,
  then all picks, both for less-than-pallet
  quantities and for full-pallet quantities, are
  from the fast-pick area.

26
Synthesizing the corresponding net-benefit
function

• Parameters
• N size of the fast-pick area (in pallet storage
  locations)
• p_i number of less-than-full-pallet picks for
  SKU i
• d_i number of pallets moved by
  less-than-full-pallet picks for SKU i
• P_i number of full-pallet picks for SKU i
• D_i number of pallets moved by full-pallet
  picks for SKU I (D_i P_i)
• ub_i maximum on-hand inventory for SKU i (in
  number of pallets)
• s savings per pick when picking from fast-pick
  area (/pick)
• c_r cost of restocking trip (/trip)
• Primary Decision variables
• x_i number of pallets from SKU i to be stored
  in the fast-pick area
• The net-benefit function for SKU i
• c_i(x_i)

0 if x_i 0
s p_i - c_r d_i if 0 lt x_i lt ub_i
s (p_iD_i) if x_i ub_i
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Optimal SKU selection and fast-pick storage
allocation

• max ?_i c_i(x_i)
• s.t.
• ?_i x_i ? N
• x_i ?? 0, 1, , ub_i , ? i

28
Plotting the net-benefit function
net benefit
s (p_iD_i)
s p_i - c_r d_i
1
2
3
4
ub_i
ub_i-1
num. of pallets
0
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Characterizing the Optimal Solution

• Theorem (The law of none or one or all) Each
  SKU that is picked from pallets should either not
  be in the fast-pick area at all or it should
  have only one pallet in it or it should have all
  of its on-hand inventory in it.
• Remark The theorem can be immediately extended
  to the case that a minimum threshold is set for
  the number of pallets from SKU i stored in the
  fast-pick area, lb_i. In that case, the three
  possibilities are 0, lb_i and ub_i.

30
Solution Algorithm

• Rank all SKUs by the marginal rate of return,
  (s p_i - c_r d_i) / lb_i, for increasing their
  presence in the fast-pick area from 0 to lb_i.
• Repeatedly,
• choose the SKU with the greatest marginal rate of
  return
• increase its amount to the next target level,
  lb_i or ub_i
• update its marginal rate of return to
• (s D_i c_r d_i) / (ub_i - lb_i) if the
  current level for SKU i is lb_i
• 0 otherwise
• until either all SKUs have a marginal return
  rate of 0, or
• for the selected SKU, the requested allocation
  exceeds the available free storage positions.
• Allocate the remaining free storage positions, in
  a way that maximizes the total benefit while
  respecting the problem constraints