a logistic management system integrating inventory...

A Logistic Management System Integrating Inventory Management

and Routing

Ana Luísa Custódio*

Dept. Mathematics

FCT – UNL

July 2002

Rui Carvalho Oliveira**

CESUR/Dept. Civil Engineering

IST – UTL

*[email protected] **[email protected]

Outline

1. Framing

2. The Case Study (Vagelpam)

3. Daily Demand

4. The Basic Model (Phase I)

5. Results (Phase I)

6. Sporadic Demand (Phase II)

7. Safety Stocks

8. Feasibility Analysis of the Solution

9. Conclusions

1. Framing

Classical ApproachesRoutingOptimization of frequency and replenishment quantities

Integrated approach of inventory management and routing

– Inventory Routing Problem (IRP) –

Strategical problem not daily management problem

2.1. The Case Study (Vagelpam)

Distribution– made from a central depot located near Lisbon– delivery points located south of Coimbra

Nestlé’s frozen products151 products321 delivery points 3937 items

FleetCapacity (pallets)

Number of vehicles

5 16 18 1

10 4


Delivery

Vagelpam

Nestlé

Order’s Database

Delivery Point Order Note

Order Note

Order Note

Stock

EmergencyOrder

Replenishment


variable transportation costdelivery fixed costinventory holding costorder fixed cost

Distance Matrixuse of VisualRoute

Cost Structure

Times involved:

transportation time – average traveling speed of 70km/h

waiting time for delivery

delivery time – related to quantity

Journey Duration- maximum of 13 hours


Orders

Quantity

01020304050607080

10 30 50 70 90 400

800

4000

1000

016

000

More

Orde r quantity (boxe s ) by de live ry point

Freq

uenc

y

0%10%20%30%40%50%60%70%80%90%100%

02468

101214161820

1003005007001700270037004700820014200

More

Or de r quantity (boxe s ) by product

Fre

qu

en

cy

0%10%20%30%40%50%60%70%80%90%100%

0

500

1000

1500

2000

2500

5 15 25 35 45 55 65

Number of orders by item

Freq

uenc

y

0%

20%

40%

60%

80%

100%

0

5

10

15

20

25

30

35

5 15 25 35 45 55 65 75 85 95More

Num ber of de livery points by product

Freq

uenc

y

0%10%20%30%40%50%60%70%80%90%100%

0

50

100

150

200

250

5 15 25 35 45 55 65 75More

Number of products ordered by delivery point

Freq

uenc

y

0%10%20%30%40%50%60%70%80%90%100%

0500

100015002000250030003500

50 200

350

500

650

800

950

3000

060

000

Order quantity (boxes) by item

Freq

uenc

y

0%10%20%30%40%50%60%70%80%90%100%

Result

Excluding 70% of items (2761 items)

Representing 5% of total quantity ordered

2.5. The Case Study (Vagelpam)Pareto Analysis on Quantity

10%

20%

70%

0

50

100

150

200

250

300

350

5 15 25 35 45 55 65

Number of records/Item

Freq

uenc

y0%10%20%30%40%50%60%70%80%90%100%

25% of items with 5 or less records

Power Law for

Variance Estimation

Daily Demand

3.1. Daily Demand

Power Law for

Variance Estimation

R² adjusted = 0,947F(1,660)=11813, p<0.000

Estimate t(660) p-levelD -0,133 -4,535 0,000P 2,165 108,687 0,000

Residuals Analysis

P2 Cµσ =

( ) ( )( )ClogD with

,µlogPDσlog 2

=×+=

Scatterplot (NEW.STA 4v*662c)

y=-0,133+2,165*x+eps

LOGMED

LOG

VAR

-8

-4

0

4

8

12

16

-3 -1 1 3 5 7

Normal Probability Plot of Residuals

Residuals

Expe

cted

Nor

mal

Val

ue

-4

-3

-2

-1

0

1

2

3

4

-3 -2 -1 0 1 2 3

3.2. Daily Demand

Sum of 4 lognormals

Expected

Variable VAR1 ; distribution: Lognormal

Kolmogorov-Smirnov d = ,0849710, p = n.s.

Chi-Square: ,5165555, df = 2, p = ,7723823 (df adjusted)

Category (upper limits)

No

of o

bs

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02468

101214161820222426

0 220 440 660 880 1100 1320 1540 1760 1980 2200

Lognormal Distribution

Expected

Variable VAR1 ; distribution: Normal

K-S d = ,0243330, p = n.s. Lilliefors p < ,15

Chi-Square: 22,43448, df = 20, p = ,3174884 (df adjusted)

Category (upper limits)

No

of o

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0102030405060708090

100110120

60 62 64 66 68 70 72 74 76 78 80 82 84 86 88 90 92 94 96 98 100

102

104

106

108

110

112

114

116

118

120

122

124

126

128

130

4.1. The Basic Model (Phase I)

Multiples Power of Two Policies

T2 – optimal replenishment period

of a multiple power of two

∈ *

*2 T2,

2TT

( )( ) 1.06TfTf

*

2

≤

{ }... 3 , , 2 , 1 , 0j, 2TT jB ∈×=

4.2. The Basic Model (Phase I)

Viswanathan and Mathur’s heuristic (1997)items allocated to clusters

different clusters replenished by distinct vehicles

allocation based on minimal replenishment period− EOQ formulae− power of two policies

Vehicles capacities − cyclic; decreasing order

TSP’s resolution:− nearest neighbor− minimal cost insertion− 2-optimal procedure

4.3. The Basic Model (Phase I)begin

create a new empty cluster

for all the items not yet allocated toclusters do

compute fixed replenishment cost of theitem in an empty cluster

for all the clusters with sufficient vehiclecapacity and available journey duration

compute fixed joint replenishment cost ofthe item in that cluster

choose the minimum fixed cost

compute T2

choose the minimum T2

create new route in the cluster and allocatethe item to the route and to the cluster

Is T2 of the new route> T2 of the last route in the

cluster ?

No merge routes compute replenishment timefor the new route

verify vehicle capacityand journey duration

Are all the itemsallocated?

Yes

endYes

Is there anempty cluster?

No

create a newempty cluster

No

Yes

Is vehicle capacityor journey duration

exceeded?

No

reduce replenishmenttime of the route

Yes

5.1. Results (Phase I)

89 clusters (75 seconds)

02468

1012141618

1 3 5 7 9 11 13

More

Jo u r n e y Du r atio n (h o u r s )

Freq

uenc

y

0%10%20%30%40%50%60%70%80%90%100%

Average vehicle occupation rate = 89%

Journey duration (including traveling, unloading and waiting times)

Average = 8.21 h

Maximum = 13.1 h

010203040506070

0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 More

Vehicle Occupation Rates

Freq

uenc

y

0%

20%

40%

60%

80%

100%


maximum of 2 routes (87% only one)average daily demand is a key factor for route constructiongeographical proximity between localities belonging to the same cluster

Vehicles

0102030405060

1 2 3 4 More

Number of Localities

Freq

uenc

y

0%20%40%60%80%100%

0

10

20

30

40

50

60

1 2 3 4 5 6 More

Number of Delivery Points

Freq

uenc

y

0%

20%

40%

60%

80%

100%

low number of visited localities− 90% visit up to two localities

low number of delivery points− 80% supply one or two delivery

points


number of products replenished variable− 58% up to 10 products− 92% up to 30 products

Vehicles

05

10152025303540

5 10 15 20 25 30 35 40 45More

Number of Products

Freq

uenc

y

0%

20%

40%

60%

80%

100%

6.1. Sporadic Demand (Phase II)

represents only 5% of total quantity ordered

sufficient available capacity to allocate the overall quantities to be delivered

Use of several heuristics

starting with items with:– the highest daily demand

– the lowest daily demand

choosing from within the routes, that serve the corresponding locality, the one with:

– the highest available capacity

– the lowest requirement

– the highest ratio available capacity/lowest requirement

6.2. Sporadic Demand (Phase II)

Best results Highest daily demand/

Lowest requirement

Only 38% of the remainder quantity allocated

Not allowed:route expansion

order splitting among routes

Daily operations management problem

7. Safety Stocks

∆ - safety stock

RT - demand during replenishment cycle Normal or Lognormal

62% less then 10 boxes

( ) ( ) 0.10R∆QP0Stock FinalP T <<−+=<

∑=

=T

1iiT RR

0100

200300

400500

600700

10 30 50 70 90 110

130

150

170

190

MoreSafety Stocks (boxes)

Freq

uenc

y

0%

20%

40%

60%

80%

100%

Future Work:Dimensioning of the safetystocks intern to model

8. Feasibility Analysis of the Solution

Heuristic Fleet

1 vehicle with routes every 2k days ⇔ 2j-k pseudo-vehicles with routes every 2j days (j ≥ k)

Total= 89

Capacity (pallets)

Number of vehicles needed

10 528 136 125 12

Capacity (pallets)

Number of vehicles needed

10 48 16 15 1

Total= 7

9. Conclusions

Feasible solution in an acceptable computational time

Key-factors for route construction:

geographical proximity

daily demand

Vehicles with:

high occupation rates

small number of distinct routes

visiting small number of different localities and delivery points

replenishing a high number of distinct products

a logistic management system integrating inventory...

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