Design of the fast-pick area
Based on Bartholdi & Hackman,
Chpt. 8
The “fast-pick” or “forward-pick” or
“primary-pick” area
Primary
picking
Restocking
Shipping
Receiving
Forward pick
Area
Reserves
picking
Reserves
Area
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 SKU’s 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.
Major issues to be resolved
• Which SKU’s 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)
Available approaches
• Fluid models (Bartholdi & Hackman, Chpt. 8):
– essentially volume-based analysis of the underlying activity
– more appropriate for SKU’s 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
A fluid model for determining the
optimal allocation of forward-pick
storage to a pre-determined set of SKU’s
• 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 ($/trip)
• Determine
– u_i: storage volume to be allocated to SKU i, i=1,…,n (cubic ft)
s.t. total restocking cost (rate) is minimized ($/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
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
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 / V^2
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.
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
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
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 s^2. Then, the above ratio is
approximated by:
m^2 + s^ 2
= 1 + CV^2
m^2
Hence, the more diverse the rates of flow of the various SKU’s,
the more sub-optimal is the performance of the two heuristics.
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 SKU’s that received less than their minimum
required space, when solving the problem without considering
these constraints.
– Increase the allocations of these deficient SKU’s 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 SKU’s.
• A similar type of logic can be applied for the
accommodation of constraints imposing a maximum
allocation
Selecting the SKU’s 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, i=1,…,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
s*p_i - c_r*(f_i / u_i) if u_i > 0
($/year)
Plotting the “net benefit” function
c_i(u_i)
(c_r*f_i) / (s*p_i) : minimum volume to be stored, if any
u_i
Problem Formulation
max _i c_i(u_i)
s.t.
_i u_i
 V
u_i  0,  i
A near-optimality condition:
The SKU’s that have the strongest claim to the fast-pick
area are those with the greatest viscocities, p_i /  f_i.
Algorithm for computing a near-optimal
solution
• Sort all SKU’s from most viscous to least (p_i /  f_i)
• For k = 0 to n (total number of SKU’s):
– Compute the optimal allocation of the fast-pick storage if it
accommodates only the first k SKU’s 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.
Proving the near-optimality of the SKU
selection algorithm
Theorem: Choosing SKU’s 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 (s*p_i)
where z* denotes the optimal objective value.
• When there are many SKU’s, the net benefit associated with a
single SKU will be a very small/negligible fraction of the
overall net benefit.
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
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_r*f_i / (s*p_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.)
Extending the basic models (cont.)
• Dealing with set-up costs:
-m_i
c_i(u_i) =
if u_i = 0
{ s p_i - (c_r f_i / u_i) - M_i if u_i > 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
Determining the Optimal Size of the
Fast-Pick Area
• Basic trade-off: A larger fast-pick area means more SKU’s
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 - b*V
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.
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 _(i=1)^k  f_i /  ( b _(i=1)^k p_i)
for some number k of the most viscous sku’s and where b
is the decreasing rate of the pick savings per volume unit
of the fast pick-area.
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 SKU’s in decreasing viscocity
• For k=0 to n, consider the set of the k most viscous SKU’s
and compute:
– the optimal storage size V^k, corresponding to this SKU selection
(e.g., V^k = V* =  c_r _(i=1)^k  f_i /  ( b _(i=1)^k p_i)
– the optimal allocation of V^k to the corresponding SKU sub-set
(e.g., for i=1 to k, u_i^k = ( f_i / _j=1^k  f_j) * V^k )
– the resulting total benefit
(e.g., _i=1^k [s(V^k) p_i - c_r f_i / u_i^k ] )
• 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 i=1 to k; 0, otherwise
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
• => 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 smallitem picking).
A typical operating policy for
case-pick-from-pallet 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.
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
s p_i - c_r d_i
s (p_i+D_i)
if x_i = 0
if 0 < x_i < ub_i
if x_i = ub_i
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
Plotting the net-benefit function
net
benefit
s (p_i+D_i)
s p_i - c_r d_i
0
1
2
3
4
ub_i-1
ub_i
num. of pallets
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.
Solution Algorithm
• Rank all SKU’s 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 SKU’s 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