Page 1
“Performance Analysis of Split-Case Sorting Systems” by Johnson and Meller
Main Points
• Split-case sorting system operation and technology.
• Use Bernoulli process to model induction process and characterize negative effect of inductor interference.
• Faster inductors should be placed upstream.• Split induction systems can outperform side-by-
side induction systems.• Presorting can be used for increased throughput
(but it can hurt picking).
Page 2
“Performance Analysis of Split-Case Sorting Systems” by Johnson and Meller
Introduction
• Used in order fulfillment centers (e.g., Amazon.com, L. L. Bean, or Sears).
• Need to fill lots of orders with common items.• Can either go out and pick each order individually
(stopping at the same location multiple times) or pick a batch of orders at a time and then sort them into individual orders.
• When order commonality is high enough, then batch picking of full cases and splitting the cases for sorting is efficient.
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Source: W&H Systems, Inc.
Page 4
•
Source: Cleco Systems, http://www.cleco.nl/objects/images/sortation.jpg
Page 5
Automated Sorting System
Scanner
Non-Recirculating Sorter
Destination Bins (1 through B/2)
Induction Station (N=2)
conveyorfrom picking area
12B/2
BB-1
Destination Bins (B/2+1 through B)
1
2
Source: Johnson, M.E. and Meller, R. D., “Performance Analysis of Split-Case Sorting System," to appear in M&SOM.
Page 6
Sorting System Sub-Systems
• Induction:– typically manual (humans)– can be automated (robots or conveyors)
• Sortation:– manual (humans) or– automated conveyors (tilt-tray, bomb-bay, cross-belt)
• Packing:– place items into shipping carton– check for all items (quality assurance)– add packing slip/invoice– button up
Page 7
Induction Area
Scanner
Parts from
Picking
Packing Station
Source: Hanover Direct, Roanoke, VA
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Induction Process
• Paper makes claim that induction process is the criticalprocess/sub-system.
• Tends to limit throughput of the system once sorter hardwareis in place.
• Decisions:
– How many inductors?
– Where to place them at stations and within station?
• Objective:
– Would like to minimize cost (function of number of pickersand picking stations).
– Need to meet throughput requirements.
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Notation
• pi = probability that inductor i can induct onto a movingconveyor (i = 1, . . . , N)
• pi = λi/s, λi < s
• λi = the induction rate of inductor i if working in isolation
• s = speed of the conveyor
• λ′i = the effective induction rate of inductor i; λ′
i ≤ λi < s
Page 10
Side-by-Side Inductor Interference
2
1
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Side-by-Side Inductors
• λ′1 = λ1 since inductor 1 is never blocked
• What about inductor 2?
– Geometric Distribution (p): Mean number of trials until first
success equals 1/p.
– Mean number of trials until inductor 2 ready to place an item on
= 1/p2 − 1.
– Mean number of trials until inductor 2 sees an empty tray =
1/(1 − p1).
– Add these together and take the inverse . . . yields the probability
that inductor 2 hits the next tray.
– Multiply by s and you have the effective induction rate of
inductor 2: λ′2 = [ 1
λ2/s− 1 + 1
1−λ1/s]−1s.
Page 12
Side-by-Side Inductor Interference
0%
20%
40%
60%
80%
100%
0% 20% 40% 60% 80% 100%
Nominal Induction Rate of Each Inductor as a % of the Conveyor Speed
Effe
ctiv
e To
tal I
nduc
tion
Rat
e as
a
% o
f the
Con
veyo
r Spe
ed
BoundSBS
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Faster Inductor
• When one inductor is faster than the other, which inductorshould be first?
• Can answer mathematically (see Result 1).
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Faster Inductor
0.0%
1.0%
2.0%
3.0%
4.0%
5.0%
6.0%
10%/90% 20%/80% 30%/70% 40%/60%% of Total Nominal Induction Rate for Inductor 1 and 2
% in
crea
se in
Eff
ectiv
e In
duct
ion
Rat
e by
Pl
acin
g th
e Fa
st In
dust
or F
irst
90%70%50%
Total Nominal Induction Rate as a
% of Conveyor
Speed
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Split Inductors
• Now both workers will experience blocking (not just the second
inductor).
• Assume that items are equally-likely to be destined for any pack
station.
• As a result, 1/2 of the items will be sorted before arriving at the
other station.
• λ′2 =
[1
λ2/s− 1 +
1
1 − λ′1/(2s)
]−1
s
• λ′1 =
[1
λ1/s− 1 +
1
1 − λ′2/(2s)
]−1
s
• λ′1 = λ′
2 = λ′ ⇒ (3)
• Result 2 tells us that SPL always does better than SBS for λ < s.
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Split Results
Result 2: For two inductors each with nominal induction rate λ
(λ < s), the total effective induction rate of a split system is larger
than that of a side-by-side configuration (Λ′SPL > Λ′
SBS).
Result 3: For two inductors working in a split configuration with
nominal induction rates limited by the conveyor speed (i.e., λi = s,
i = 1, 2), the total effective induction rate is expressed as
Λ′SPL =
(4
3
)s.
Result 4: For N inductors working in an equally spaced split
configuration (with B > N evenly distributed between the
inductors) with nominal induction rates limited by the conveyor
speed (i.e., λi = s, i = 1, . . . , N), the total effective induction rate is
expressed as Λ′SPL =
(2N
N+1
)s. Moreover, limN→∞ Λ′
SPL = 2s.
Page 18
Improvement with Split
0%20%40%60%80%
100%120%140%160%180%200%
0% 20% 40% 60% 80% 100%
Nominal Induction Rate of Each Inductor as a % of the Conveyor Speed
Effe
ctiv
e To
tal I
nduc
tion
Rat
e as
a
% o
f the
Con
veyo
r Spe
ed
BoundSPLSBS
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Presorting to Improve Sorter Throughput
• With two stations and no presorting:
λ′ =[
1λ/s
− 1 +1
1 − λ′/(2s)
]−1
s
• With two stations and presorting that leads to dropoffprobability equal to d (d > 0.5):
λ′ =[
1λ/s
− 1 +1
1 − (1 − d)λ′/s
]−1
s
• Note that this improves sorter throughput at the price ofdecreasing picking throughput.
Page 20
Presorting to Improve Sorter Throughput
40%
60%
80%
100%
120%
140%
160%
180%
200%
50% 60% 70% 80% 90% 100%
Percent Drop-Off Before Next Station
Effe
ctiv
e To
tal I
nduc
tion
Rat
e as
a
% o
f the
Con
veyo
r Spe
ed
100%
75%
50%
25%
Inductor Nominal Rate
as a % of Conveyor
Speed
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Approximate Model for Low Induction Variance
• Motivated by case where inductors found a rhythm.
• Did not see such “random” blocking as model would predict.
• Used an approximate queueing model based on lower boundand upper bound of throughput.
• Lower bound on throughput: Geometric Model.
• Upper bound on throughput: Finite Model.
• See paper for details (pp. 267–269).
• Approximation performed well (see Tables 1–4).
Page 22
JOHNSON AND MELLERPerformance Analysis of Split-Case Sorting Systems
MANUFACTURING & SERVICE OPERATIONS MANAGEMENT
Vol. 4, No. 4, Fall 2002 271
Table 1 Results for Two Side-by-Side Inductors
SimulationExperiment
AVG TBA
Inductor
1 2Total
Intensity
VAR TBA
Inductor
1 2
Total Induction
GeometricEstimate
(�G)
FiniteEstimate
(�F )
Approx.Estimate
(�A)SimulationEstimate
SimulatedHalf-Width
95% % Error
12345
55555
55555
40.0%40.0%40.0%40.0%40.0%
0.801.302.403.203.60
0.801.302.403.203.60
39.0539.0539.0539.0539.05
39.2339.2339.2339.2339.23
39.1939.1939.1939.1839.18
39.3039.2339.2139.2139.21
0.010.020.030.030.03
0.27%0.10%0.07%0.07%0.06%
6789
10
44444
44444
50.0%50.0%50.0%50.0%50.0%
0.751.502.102.552.78
0.751.502.102.552.78
48.0848.0848.0848.0848.08
48.5348.5348.5348.5348.53
48.4148.4048.3948.3848.37
48.6048.5048.4648.4548.42
0.020.040.030.040.04
0.40%0.21%0.15%0.14%0.08%
1112131415
33333
44444
58.3%58.3%58.3%58.3%58.3%
1.001.331.601.801.90
0.751.502.102.552.78
55.5555.5555.5555.5555.55
56.4156.4156.4156.4156.41
56.0956.0756.0556.0456.03
56.3356.2456.1956.1556.08
0.030.030.040.040.05
0.41%0.30%0.25%0.20%0.08%
1617181920
33333
33333
66.7%66.7%66.7%66.7%66.7%
1.001.331.601.801.90
1.001.331.601.801.90
61.9061.9061.9061.9061.90
63.3363.3363.3363.3363.33
62.7662.7362.6962.6762.66
63.0862.9462.8662.8162.75
0.030.040.040.050.05
0.50%0.34%0.27%0.22%0.15%
2122232425
22222
44444
75.0%75.0%75.0%75.0%75.0%
0.750.830.900.950.98
0.751.502.102.552.78
70.0070.0070.0070.0070.00
72.2272.2272.2272.2272.22
71.0370.9870.9370.9070.88
71.1471.0670.9470.8770.86
0.030.050.040.060.06
0.16%0.12%0.01%
�0.05%�0.03%
2627282930
22222
33333
83.3%83.3%83.3%83.3%83.3%
0.750.830.900.950.98
1.001.331.601.801.90
75.0075.0075.0075.0075.00
78.5778.5778.5778.5778.57
76.4976.4276.3576.3176.28
76.7676.5676.4476.3076.26
0.040.050.040.050.06
0.35%0.19%0.11%
�0.01%�0.02%
3132333435
22222
22222
100.0%100.0%100.0%100.0%100.0%
0.750.830.900.950.98
0.750.830.900.950.98
83.3383.3383.3383.3383.33
90.0090.0090.0090.0090.00
85.6885.5585.4585.3785.32
85.7985.5185.3185.1285.06
0.030.050.050.050.05
0.13%�0.05%�0.16%�0.29%�0.31%
TBA � Trays Between Attempts. Average % Error � 0.13%.Average (Absolute) % Error � 0.18%.
Page 23
JOHNSON AND MELLERPerformance Analysis of Split-Case Sorting Systems
MANUFACTURING & SERVICE OPERATIONS MANAGEMENT
Vol. 4, No. 4, Fall 2002272
Table 2 Results for Two Split Inductors
SimulationExperiment
AVG TBA
Inductor
1 2Total
Intensity
VAR TBA
Inductor
1 2
Total Induction
GeometricEstimate
(�G)
FiniteEstimate
(�F )
Approx.Estimate
(�A)SimulationEstimate
SimulatedHalf-Width
95% % Error
12345
55555
55555
40.0%40.0%40.0%40.0%40.0%
0.801.302.403.203.60
0.801.302.403.203.60
39.1539.1539.1539.1539.15
39.2339.2339.2339.2339.23
39.2139.2139.2139.2139.21
39.2439.2139.2339.2239.21
0.010.020.030.040.03
0.06%0.00%0.04%0.02%0.01%
6789
10
44444
44444
50.0%50.0%50.0%50.0%50.0%
0.751.502.102.552.78
0.751.502.102.552.78
48.3448.3448.3448.3448.34
48.5348.5348.5348.5348.53
48.4848.4748.4748.4648.46
48.5448.5248.5048.4948.48
0.020.030.030.030.04
0.14%0.09%0.06%0.06%0.03%
1112131415
33333
44444
58.3%58.3%58.3%58.3%58.3%
1.001.331.601.801.90
0.751.502.102.552.78
55.7655.7655.7655.7655.76
56.0956.0956.0956.0956.09
55.9855.9755.9655.9655.96
56.0256.0056.0156.0055.96
0.030.030.040.040.05
0.07%0.05%0.08%0.07%0.01%
1617181920
33333
33333
66.7%66.7%66.7%66.7%66.7%
1.001.331.601.801.90
1.001.331.601.801.90
62.7762.7762.7762.7762.77
63.3263.3263.3263.3263.32
63.1036.0963.0863.0763.06
63.2063.1763.1263.1263.09
0.030.040.040.050.05
0.16%0.12%0.07%0.08%0.04%
2122232425
22222
44444
75.0%75.0%75.0%75.0%75.0%
0.750.830.900.950.98
0.751.502.102.552.78
70.1470.1470.1470.1470.14
70.8270.8270.8270.8270.82
70.5370.5170.5070.4970.48
70.5170.4970.4570.4470.41
0.050.040.040.060.04
�0.02%�0.02%�0.06%�0.07%�0.10%
2627282930
22222
33333
83.3%83.3%83.3%83.3%83.3%
0.750.830.900.950.98
1.001.331.601.801.90
76.2276.2276.2276.2276.22
77.3577.3577.3577.3577.35
76.7776.7576.7376.7176.70
76.8076.7476.7276.6776.67
0.040.050.050.060.07
0.04%�0.01%�0.01%�0.05%�0.04%
3132333435
22222
22222
100.0%100.0%100.0%100.0%100.0%
0.750.830.900.950.98
0.750.830.900.950.98
87.6987.6987.6987.6987.69
89.9089.9089.9089.9089.90
88.4788.4288.3988.3688.35
88.4988.4188.3688.3088.29
0.040.060.060.060.07
0.02%�0.02%�0.03%�0.07%�0.07%
TBA � Trays Between Attempts. Average % Error � 0.02%.Average (Absolute) % Error � 0.05%.
Page 24
JOHNSON AND MELLERPerformance Analysis of Split-Case Sorting Systems
MANUFACTURING & SERVICE OPERATIONS MANAGEMENT
Vol. 4, No. 4, Fall 2002 273
Table 3 Results for Three Side-by-Side Inductors
SimulationExperiment
AVG TBA
Inductor
1 2 3Total
Intensity
VAR TBA
Inductor
1 2 3
Total Induction
GeometricEstimate
(�G)
FiniteEstimate
(�F )
Approx.Estimate
(�A)SimulationEstimate
SimulatedHalf-Width
95% % Error
12345
55555
55555
55555
60%60%60%60%60%
0.801.302.403.203.60
0.801.302.403.203.60
0.801.302.403.203.60
56.7856.7856.7856.7856.78
57.7757.7757.7757.7757.77
57.3957.3857.3657.3557.34
57.6857.4457.3257.2757.24
0.020.020.020.030.04
0.50%0.09%
�0.08%�0.13%�0.18%
6789
10
44444
44444
44444
75%75%75%75%75%
0.751.502.102.552.78
0.751.502.102.552.78
0.751.502.102.552.78
68.3868.3868.3868.3868.38
70.8370.8370.8370.8370.83
69.6669.6169.5669.5269.50
70.2869.7469.5469.4569.35
0.020.020.030.040.04
0.88%0.20%
�0.03%�0.11%�0.22%
1112131415
33333
33333
33333
100%100%100%100%100%
1.001.331.601.801.90
1.001.331.601.801.90
1.001.331.601.801.90
83.5283.5383.5283.5283.52
90.8590.8590.8590.8590.85
86.0785.9485.8385.7585.71
86.7686.1385.7485.4985.38
0.040.040.050.040.06
0.79%0.22%
�0.11%�0.30%�0.39%
1617181920
22222
22222
22222
150%150%150%150%150%
0.750.830.900.950.98
0.750.830.900.950.98
0.750.830.900.950.98
97.6297.6297.6297.6297.62
100.00100.00100.00100.00100.00
99.9699.8499.7399.6599.60
99.1098.8798.6998.5798.57
0.050.040.040.050.05
�0.87%�0.98%�1.06%�1.10%�1.05%
TBA � Trays Between Attempts. Average % Error � �0.20%.Average (Absolute) % Error � 0.46%.
Table 4 Results for Four Side-by-Side Inductors
SimulationExperiment
AVG TBA
Inductors
1–4Total
Intensity
VAR TBA
Inductors
1–4
Total Induction
GeometricEstimate
(�G)
FiniteEstimate
(�F )
Approx.Estimate
(�A)SimulationEstimate
SimulatedHalf-Width
95% % Error
12345
55555
80%80%80%80%80%
0.801.302.403.203.60
72.6272.6272.6272.6272.62
75.7075.7075.7075.7075.70
74.0574.0273.9673.9273.90
74.8274.2273.8073.6373.56
0.020.030.030.040.04
1.03%0.27%
�0.22%�0.39%�0.45%
6789
10
44444
100%100%100%100%100%
0.751.502.102.552.78
84.6184.6184.6184.6184.61
92.0393.0692.0692.0692.06
87.0786.9586.8486.7686.72
88.9087.3686.7486.4286.29
0.030.030.030.040.05
2.06%0.47%
�0.11%�0.39%�0.50%
1112131415
33333
133%133%133%133%133%
1.001.331.601.801.90
95.9295.9295.9295.9295.52
100.00100.00100.00100.00100.00
98.4798.3398.2398.1498.10
98.8798.1297.6597.3997.25
0.010.020.030.020.03
0.41%�0.22%�0.59%�0.77%�0.88%
TBA � Trays Between Attempts. Average % Error � �0.02%.Average (Absolute) % Error � 0.58%.
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ISE 5244 Johnson & Meller Sorter Paper 9'
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Conclusions
• Concepts:
– When does it pay to split induction stations?
– When does it make sense to presort the items?
– When do you need to use approximate model (with queueing
approximation)?
• Skills:
– Calculate the throughput of a side-by-side system with 2 inductors (see
pg. 12).
– Calculate the throughput of a split system with 2 inductors (see pg. 12).
– Calculate the maximum throughput of a system with N induction stations.
– Calculate the throughput of a split system with 2 inductors and presorting
(see pp. 15–16).
• Extension:
– Alluded to more than two inductors . . . how would you modify the models?