Download - “Performance Analysis of Split-Case Sorting Systems” by ...Automated Sorting System Scanner Non-Recirculating Sorter Destination Bins (1 through B/2) Induction Station (N=2) conveyor

“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).

“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.

Source: W&H Systems, Inc.

•

Source: Cleco Systems, http://www.cleco.nl/objects/images/sortation.jpg

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.

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

Induction Area

Scanner

Parts from

Picking

Packing Station

Source: Hanover Direct, Roanoke, VA

ISE 5244 Johnson & Meller Sorter Paper 1'

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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

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.

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).

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

Split Inductors


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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.

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.

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).

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%.



Vol. 4, No. 4, Fall 2002272

Table 2 Results for Two Split Inductors


AVG TBA

Inductor

1 2Total

Intensity

VAR TBA

Inductor

1 2

Total Induction

GeometricEstimate

(�G)

FiniteEstimate

(�F )

Approx.Estimate


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%.



Vol. 4, No. 4, Fall 2002 273

Table 3 Results for Three Side-by-Side Inductors


AVG TBA

Inductor

1 2 3Total

Intensity

VAR TBA

Inductor

1 2 3

Total Induction

GeometricEstimate

(�G)

FiniteEstimate

(�F )

Approx.Estimate


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


AVG TBA

Inductors

1–4Total

Intensity

VAR TBA

Inductors

1–4

Total Induction

GeometricEstimate

(�G)

FiniteEstimate

(�F )

Approx.Estimate


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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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?