Transcript Install/Qual scheduling is critical for Intel to achieve

Integrated Consolidation Facility Location
and Inventory Routing for Supply Networks
Ronald G. Askin
[email protected]
with thanks to Mingjun Xia
School of Computing, Informatics, and
Decision Systems Engineering
Arizona State University
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Overview
• Introduction
• Problem Definition and Background
• Multi-Product Integrated Supply Chain Network Model
– Multi-product FLP with Approximated IRC Function
– Inventory Routing Problem
– Integrated Problem’s Results and Analysis
• Consolidation Facility Location & Demand Allocation
• Global Sourcing Options for Multistage Production
• Conclusion and Future Work
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Who’s That Speaker?
Ronald G. Askin, Director
School of Computing, Informatics, and
Decision Systems Engineering
Arizona State University
Tempe, AZ 85287-8809 USA
[email protected]
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Arizona State University
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The State of Arizona
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Background and Activities
• BS in IE, Lehigh University (in the time of punch cards)
• MS OR and PhD in Industrial and Systems Engineering, Georgia Tech
• Professor of Industrial Engineering
• Fellow of Institute of Industrial Engineers (IIE)
• Former IIE Board of Trustees Member
• Former Chair of Council of Industrial Engineering Academic Dept Heads
• Former President of INFORMS M&SOM Society
• Editor-in-Chief IIE Transactions
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IIE Transactions
The flagship journal of the
Institute of Industrial Engineers and
hopefully your preferred choice for
publication.
http://www.tandfonline.com/toc/uiie20/current
• Methodological focus in most papers
• Real world applications/impact
• Original, innovative contribution required
• Novel problems and models encouraged
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Logistics: Facility Layout to Supply Networks
•
•
•
Production Control Systems
Supply Chain Design
Batch Sizing/Lot Streaming
• Queuing Networks
• Material Flow & Capacity Models
• Facility Layout
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Introduction
•
Supply Chain Management (SCM)
The goal: To deliver the right
product to the right place at the
right time for the right price,
while minimizing system-wide
costs and satisfying service
requirements.
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Supply Chain Management Decisions
Strategic Level
Tactical Level
Long Term
• Corporate objectives
• Capacity / Facilities
• Markets to operate
• Location
• Resources
Medium Term
• Aggregate planning
• Resource allocation
• Capacity allocation
• Distribution
• Inventory management
• Shop floor scheduling
Operational Level
• Delivery scheduling
• Truck routing
Near Term
Facility Location
Problem (FLP)
Inventory Control
Problem (ICP)
Vehicle Routing
Problem (VRP)
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Motivation
•
•
•
Lots of research has been done in each area in SCM, but few models
comprehensively address the integrated network.
To achieve a global optimal (or near optimal) solution, it is necessary to
consider the entire system in an integrated fashion and include all trade-offs in
a realistic fashion.
We will look at the Distribution side (post production).
Inventory Strategy
• Forecasting
• Inventory decisions
• Purchasing and supply
scheduling decisions
• Storage fundamentals
• Storage decisions
Transportation Strategy
Customer
service goals
• Transport fundamentals
• Transport decisions
Location Strategy
• Location decisions
• The network planning process
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Our Distribution Problem
Made Here in Volume
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Distribution Problem Scenario
Product Mixes Sold here by the Item at many Outlets
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Global Reality – But let’s start regionally
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Our Distribution Problem
•Assume (global) manufacturing system is defined.
•Goal: Distribute completed products to retail outlets.
•Assume goal is a (distribution) system optimal solution.
•Assume a relatively stable environment.
•Assume system to be designed from scratch – (any
existing facilities could be sold for value or are on shortterm leases).
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Planning Decisions
Where to place Distribution Centers?
How large to make DCs?
How to ship from Factory to DC – Quantity, frequency, form, mode?
How to take advantage of load consolidation opportunities?
How to serve each retail outlet – from where and how often?
How much safety inventory to keep and where to keep it?
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What’s Relevant?
•Locations of Producers
What Else?
•Locations of Retailers
•Cost of Transportation
•Cost of Facilities by site/capacity(Fixed, Variable Operating)
•Vehicle Capacities
•Demands and Patterns
•Product Substitutability
•Inventory and Shortage Costs/Policies
•Product Lifetime
•Supply Dependability and Lead Times
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Important (Real-World) Factors Ignored
Insert your list here:
• Stochasticity of demand
• Dynamic nature of demand (multiple periods)
• Substitutability of products
• Strategic corporate initiatives (profit, service, competitiveness)
• Financial risk and return on investment
• Taxes, duties, exchange rates if multinational
• Reverse logistics (collection, refurbishment)
• Direct shipments
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Facility Location Problem (FLP)
•
Fermat-Weber (1909): A simple facility location problem in which a single
facility is to be placed, with the only optimization criterion being the
minimization of the sum of distances from a given set of point sites.
•
More complex problems: the placement of multiple facilities, constraints on
the locations of facilities, and more complex optimization criteria.
•
The goal: to pick a subset of potential facilities to open, to minimize the sum of
distances from each demand point to its nearest facility, plus the sum of fixed
opening costs of the facilities.
•
The facility location problem on general graphs is NP-hard to solve optimally,
by reduction from (for example) the Set Cover problem.
•
Daskin (2002), Ozsen (2008): include inventory control decisions in FLP.
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Inventory Control Problem (ICP)
•
•
Harris (1913): Economic order quantity (EOQ)
Clark and Scarf (1960): Multi-echelon Inventory
•
Inventory control: the supervision of supply, storage and accessibility of items
in order to ensure an adequate supply without excessive oversupply.
– Where to hold inventory?
– When to order?
– How much to order each time?
•
The goal: the order quantity and the reorder point are determined such that the
total cost is minimized.
– Total cost = Purchasing cost + Setup Cost + Holding Cost + Shortage Costs
•
The single-item stochastic inventory control problem is NP-hard even in the
case of linear procurement and holding costs. (Halman et al. , 2009)
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Vehicle Routing Problem (VRP)
•
Dantzig and Ramser (1959):To deliver goods located at a central depot to
customers who have placed orders for such goods.
•
The goal: to minimize the cost of distributing the goods.
•
The vehicle routing problem in general is NP-hard as it lies at the intersection of
these two NP-hard problems:
– Traveling Salesman Problem
– Bin Packing Problem
•
Inventory Routing Problem (IRP): An extension to include inventory concerns.
Kleywegt, Nori, Salvesbergh, Transportation Science, 2002
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How Good are the Models?
“A conclusion that can be drawn from the literature devoted to the
uncapacitated facility location problem and its extensions is that
the research field has somehow evolved without really taking the
SCM context into account. Features … have been included in the
models in a rather general way and specific aspects, that are
crucial to SCM, were disregarded. In fact, extensions seem to
have been mostly guided by solution methods.”
- Melo et al. EJOR, 2009
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Principles to Keep in Mind
1. Pooling Synergy
Safety Stock
Di
Z   N  Var ( Di ) 
1/2
Di
Di
Assumes independence
Di
Z Var ( Di )1/2
Di
Z Var ( Di )1/2
Di
Z Var ( Di )1/2
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Principles to Keep in Mind
2. Inventory vs. Service Level
What’s the Traditional Perspective?
100%
Fill Rate
Inventory
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Comment on Second Principle:
Little’s Law
In Steady State,
Average Inventory = Consumption Rate x Ave. Time in System
N = XT or L = λW
Diminishing Returns: Beyond the “elbow” more
inventory is just more cost and more opportunity for
degradation, loss, congestion and cost!
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But it’s even worse
Beyond a threshold increasing inventory reduces sales!
• Congestion slows service response
• Inventory is outdated
• Forecast horizons too long for accuracy
Carburetors vs. Fuel Injection
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Empirical Profile:
Know When Enough is Enough
Little's Law and Chaos
12
Remember
L =λW
10
N = XT
Throughput
8
6
Deterministic
Exponential
4
Empirical
2
0
0
10
20
30
50
40
WIP
60
70
80
90
In theory, there’s no
difference between
theory and practice,
in practice there is.
– Yogi Berra
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Multi-Product Integrated Supply
Chain Network Model
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Existing Research is Helpful but Not the Same
•
Shen, Z.J.M, Qi, L., 2007. Incorporating inventory and routing costs in strategic
location models. European Journal of Operational Research 179, 372-389.
– Single Producing Plant (One Supplier, One Product)
– Uniformly located customers across an area
– (Q,r) inventory ordering/replenishment model for DC
– Fixed and identical routing frequency from DC to customers
– Single routing tour from each DC (1 vehicle) for cost estimation
– Uncapacitated DCs
•
Javid, A.A., Azard, N., 2010. Incorporating location, routing and inventory
decisions in supply chain network design. Transportation Research Part E 46,
582-597.
– Single product, no transhipment
– (Q,r) inventory model for DC
– Known delivery route frequency
– Fixed vehicle capacity per year
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Our Problem (Model)
Integrated supply chain network design:
location, transportation, routing and inventory decisions
Multi-product and plant supply chain network.
Transshipments between DCs.
Non-uniformed (clustered) customer locations.
Multiple routes with model-determined frequencies from DCs.
Nonlinear inventory costs (safety stock).
Full truck load deliveries to DCs with choice of truck size.
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Consolidation and Transportation Centers
Retailer
Consolidation
Center
Distribution
Center
Facility
A: Through Consolidation Center
Fixed Location Cost
V.S.
Transportation savings
Easy management
Retailer
Retailer
Distribution
Center
Facility
Facility
B: Only Distribution Center
C: Point-to-Point Transportation
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Multi-Product Integrated Supply Chain Network
Model
1
1
2
2
3
1
3
1
4
2
5
3
1
2
5
2
3
6
4
6
4
7
3
5
8
5
6
9
6
4
7
8
9
4
Facility
Consolidation
Distribution
10
10
11
11
Retailer
Facility
DC
Retailer
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Problem Description
1
2
3
1
1
2
4
5
2
3
6
4
3
5
7
8
6
9
4
10
– Location decision: How many DCs
to locate, where to locate, how much
capacity at each opened site.
– Transportation decision: Allocate
facilities and retailers to opened DCs.
– Routing decision: Routing tours and
frequencies to retailers.
– Inventory decision: How often to
reorder, what level of inventory stock
to maintain.
11
DC
Facility
Retailer
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Problem Description
1
2
3
1
1
2
4
5
2
3
6
4
3
5
7
8
6
9
4
10
11
Facility
DC
Facility – DC – Retailer:
– Each production facility supplies a single
product.
– Retailers are clustered in the service region.
– Demand follows a known stationary
distribution.
– Single source: all products at one retailer
should be delivered by one DC.
– Full truck load (FTL) shipping is used from
plants to DCs and between DCs, multiple
truck size choices exist.
– Routing delivery is used for shipment from
DCs to retailers.
Retailer
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Research Scope and Activities
Two-phase:
Phase I: Multi-product FLP with Approximated IRC Function
• DC Locations and plant/retailer assignments
– Approximate cost function for routing delivery cost (Shen, Z.J.M, Qi, L., 2007).
Phase II: Inventory Routing Problem
• Routing tours and frequencies
– Solve the routing problem for each open DC and retailers assigned to it.
1
1
1
2
2
2
3
1
3
1
1
2
5
2
3
6
3
4
2
5
1
2
6
4
6
4
3
7
3
5
8
5
8
5
6
9
6
9
6
4
Facility
2
3
7
DC
9
4
DC
10
7
8
4
Facility
4
5
2
3
4
3
1
1
4
10
11
11
Retailer
Retailer
10
11
Facility
DC
Retailer
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Cost Components
FC: Annualized fixed cost of locating DCs
1
2
3
Qpj
1
1
4
2
5
2
3
Qpj’j
4
6
3
7
5
8
6
9
4
10
11
Facility
p
DC
j, j’
FC   jJ ,kK f jk O jk
SC: FTL Shipping cost from plants to DCs
and between DCs
Q = total shipped/time
q = quantity per trip (shipping mode capacity)
A = cost per trip
T = if use that truck size
Q pj ' j   iI  piYpj ' ji
Q pj   j 'J ,iI  piYpjj ' i
q pj   lL ql Tpjl
Apj   lL  a pjl  bpjl ql  Tpjl
Retailer
i
q j ' j   lL ql T j ' jl
SC j   pP Apj
Q pj
q pj
Aj ' j   lL  a j ' jl  b j ' jl ql  T j ' jl
  j 'J , j ' j Aj ' j

pP
Qpj ' j
qj' j
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Cost Components
1
SSC: Safety stock inventory holding cost at DCs
2
3
Qpj
1
1
2
SS pj  z

lt pj iI , j 'J  pi2 Ypjj 'i   j 'J , j ' j lt j ' j iI  pi2 Ypj ' ji

4
5
2
3
Qpj’j
4
6
3
5
7
8
6
RIC: Regular inventory holding cost at DCs
9
RI pj 
4
q pj
2v p
  j 'J , j ' j
qj' j
2v p

Q pj ' j
pP
Q pj ' j
10
Ypjj’I binary route indicator variables
11
Facility
DC
Retailer
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Phase I: Multi-product FLP with
Approximated IRC Function
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Approximated IRC Function
IRC: Annual inventory routing cost from DCs to retailers
rji a computationally estimated parameter to represent the annual
IRC at retailer i if assigned to DC j
IRC   iI , jJ rji y ji
Routing
cost
 a  c ji
 ji  min 
nN 
 n ji
  pi

 z  pi
 f n   pP hpi 
 2 fn


Inventory cost
1  ji  


fn
s  

1
2
12
3
11
n ji  max | R j (i) |
s.t.:
 ji   ( l , m )A ( i ) dlm  D
j

mR j ( i ), pP
 pm v p
 ji
4
10
DC
6
9
5
7
q
8
Nearest insertion
method
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Approximated IRC Function
IRC: Annual inventory routing cost from DCs to retailers
rji a computationally estimated parameter to represent the annual
IRC at retailer i if assigned to DC j
IRC   iI , jJ rji y ji
direct shipping method

  pi
1 2d ji  
 ji  min  a  2cd ji  f n   pP hpi 
 z  pi


nN

 ji
s  

 2 ji

rji   ji   ji  / 2
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Problem Formulation
Minimize (FC + IRC + SC + SSC + RIC)
Subject to:

jJ


Y ji  1
j ', jJ
Ypj ' ji  1
pP , j 'J



i  I
i  I , p  P
Ypj ' ji  MY ji
i  I , j  J
Y  MW pj
iI , j 'J pjj ' i
jJ
kK



ll
W pj  PW p
O jk  1
p  P
Maximum number of PWs
Single capacity level
j  J
p  P, j  J
Tjj 'l  1
j, j '  J , j  j '
pP ,iI
Link variables
p  P, j  J
Tpjl  1
lL
Single source
Single path
Truck size selection
 piY ji   pP ,iI , j 'J , j ' j  piYpjj 'i   kK C jk O jk
Ojk ,Yji ,Ypj ' ji ,Wpj ,Tpjl ,Tjj 'l {0,1}
j  J
Throughput Capacity limit
i  I , j, j '  J , p  P, k  K
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Single Plant Warehouse Case (n = 1)
Optimal truck size from plants to PWs and transshipment between
DCs
DSRIC: direct shipping and regular working inventory holding cost

 pi
ql 



iI
DSRIC pj  min  a pjl  bpjl ql 
  pP hpj 
l
ql
2



TSRIC: Transshipment and regular working inventory holding cost
TSRIC j ' j

q
Q


pP pj ' j
 min  a j ' jl  b j ' jl ql 
  pP hpj  l
l
2
ql




Qpj ' j
pP
Qpj ' j





Shared transhipment loads
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Nonlinear Terms
Safety stock
lt pj iI , j 'J  pi2 Ypjj 'i , lt j ' j iI  pi2 Ypj ' ji
SS pj 
z2 lt pj  iI , j 'J  pi2 Ypjj 'i
S pj
S pj  z lt pj  iI , j 'J  pi2 Ypjj 'i
S pj ' j  z lt j ' j  iI  pi2 Ypj ' ji
Working inventory

Qpj ' j
pP
  j 'J , j ' j
z2 lt j ' j  iI  pi2 Ypj ' ji
S pj ' j
Recursive procedure to
update safety stock
parameters, Gebennini
et al. (2009)
Same holding rate
Qpj ' j
Qpj ' j   iI  piYpj ' ji
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Tabu Search-Simulated Annealing (TS-SA)
Tabu search can avoid search cycling by systematically preventing moves
that generate the solutions previously visited in the solution space.
Simulated annealing allows the search to proceed to a neighboring state
even if the move causes the value of the objective function to become
worse, and this allows it to prevent falling in local optimum traps.
Construction stage
•
•
•
Greedy method
Minimizing initial Fixed Cost (FC)
Minimizing initial Inventory Routing Cost (IRC)
Improvement stage
•
•
Location improvement
• Close an opened DC; Open a closed DC
Assignment improvement
• Assign one retailer to another reachable DC
• Assign one PW to another opened DC
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Ad-Hoc Heuristics
Fixed Cost (FC) and Inventory Routing Cost (IRC) are two major cost
components
FC 24% Vs. IRC 50% (Shen and Qi, 2007)
FC Heuristic: Minimizing initial FC
• Set covering problem: minimizing total number of DCs
• To include cost consideration
 Open all necessary DCs

Open additional DCs to save total cost
Improvement stage: TS-SA
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Ad-Hoc Heuristics
Inventory Routing Cost (IRC) is a major cost component
IRC accounts for 50% of total (Shen and Qi, 2007)
IRC Heuristic: Minimizing initial IRC


Open all DCs and assign retailers to its nearest DC
Close unnecessary DCs to save total cost
Improvement stage: TS-SA
Nothing new, sounds like variable selection in regression.
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Lower Bound: Without considering transshipment between DCs.

jJ
SC j   pP , jJ hpj  SS pj  RI pj 

Q pj ' j 
Q pj

pP
+
=   pP Apj
  j 'J , j ' j Aj ' j


q pj
qj' j



 q pj
q j' j
2
2


h
z
lt

Y

lt

Y


 pP, jJ pj   pj  iI , j 'J pi pjj 'i  j 'J , j ' j j ' j  iI pi pj ' ji  2  j 'J , j ' j 2


Q
q 

  pP Apj pj + pP , jJ hpj  z lt pj  iI , j 'J  pi2 Ypjj 'i  pj 
q pj
2 




hpj q pj
  pi  h z lt
2
=  pP , jJ  Apj iI


pj 
pj  iI , j 'J
pi

q
2
pj

  pP , jJ  pjW pj

 W pj


Q pj ' j
pP
Q pj ' j




n =1

 pi
hpj ql 

2
iI
Where  pj  min lL  a pjl  bpjl ql 
 hpj z lt pj  iI , j 'J  pi 

q
2

l

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Parameter Settings
•
•
•
•
•
All points (plants, DCs, and customers) are geographically dispersed in a 500 *
500 miles region.
Plants are randomly distributed, retailers are clustered into m groups with the
centers of gravity also randomly distributed in this space.
8 different data sets with each set including 15 scenarios with sizes ranging
from 20 to 200 retailers and 5 to 20 products.
Data sets differ in fixed location cost rate (low, high), demand rate (case 1,
case 2) and holding cost rate (low, high).
All the computational times are obtained on a Intel(R) Core(TM)2 T5550 at
1.83 GHz using Windows 7. Three introduced heuristics are applied in
Microsoft Visio Studio C++. IBM ILOG CPLEX Optimization Studio is used
to solve the modified model and lower bound model.
Retailers
Case 1
Case 2
High demand
10% consume 27% TD
10% consume 80% TD
Medium demand
80% consume 70% TD
10% consume 10% TD
Low48demand
10% consume 3% TD
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TD2013
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Partial Results
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Partial Results
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Results Comparison
6.3
6.1
5.9
5.7
5.5
5.3
5.1
4.9
4.7
4.5
0
5
10
15
20
25
Greedy
30
35
40
TSSA
45
50
55
60
IRC-TSSA
65
70
75
FC-TSSA
80
85
90
95 100 105 110 115 120
LowerBound
Log of Cost (to show separation)
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x 100000
Large Problems 200 Retailers, 20 DCs
16
14
12
10
8
6
4
2
0
5
10
15
20
25
30
Greedy
35
40
TSSA
45
50
55
IRC-TSSA
60
65
70
FC-TSSA
75
80
85
90
95 100 105 110 115 120
LowerBound
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Results and Analysis
•
Computational time: Maximum scenario: 1863 seconds by heuristics V.S. No
solution after 5 hours of computation for even some small instances.
•
Objective values: the greedy solution’s objective value is reduced by 29.1%
on average ( the improvement is greater under large instances). 12.1% higher
than the lower bound (do not include transshipment consideration and large
instances do not converge completely in CPLEX).
•
IRC and FC heuristics: better than simple TSSA method, especially in large
instances, indicating the importance of a good starting point.
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Results and Analysis
•
IRC heuristic performs the best in both the number of best solution scenarios
and the average GAP.
TSSA
IRC
FC
Best Solution
Scenarios
34 of 120
66 of 120
47 of 120
Average GAP
11.6%
2.0%
3.1%
Cost %
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Results and Analysis
100%
90%
80%
70%
60%
50%
40%
30%
20%
10%
0%
13.7%
12.9%
11.7%
10.9%
6.9%
7.4%
5.4%
6.9%
57.4%
63.4%
47.1%
54.3%
21.9%
16.2%
35.9%
27.9%
FCRate =
FCRate =
Low; HCRate Low; HCRate
= Low
= High
FCRate =
High;
HCRate =
Low
SC
IC
IRC
FC
FCRate =
High;
HCRate =
High
FC: fixed location cost; IRC: inventory routing cost;
IC: regular inventory and safety stock cost at DCs; SC: total shipping cost
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Phase II: Inventory Routing
Problem
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Phase II: Inventory Routing Problem
Inventory Routing Problem (IRP)
•
Inventory management and
transportation are two of the key
logistical drivers of supply chain
management.
•
Bottom problem of the integrated
supply chain network design.
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Literature Review
•
The classification criteria used for the IRP
Characteristic
Time
Demand
Topology
Routing
Inventory
Fleet composition
Fleet size
Products
•
Alternatives
Instant
Finite
Infinite
Stochastic
Deterministic
One-to-one
One-to-many
Many-to-many
Direct
Multiple
Continuous
Fixed
Stock-out
Lost sale
Homogeneous Heterogeneous
Single
Multiple
Unconstrained
Single
Multiple
Back-order
Missing/Rare in IRP research
– Uncertain demands at retailers
– Variable routing frequency
– Optimal number of vehicles
– Nonlinear characteristics of routing cost and lead time
……
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Problem Description
•
Inventory routing problem
– One distribution center (DC) and multiple retailers (R).
– Each retailer has an independent demand for the product, follows normal
distribution.
– The DC uses homogeneous capacitated vehicles for routing delivery.
– Routing frequencies fall in a discrete set such as daily, every other day,
etc.
•
Decisions
– Routing tour to each retailer.
– Routing frequency of each tour.
•
Total cost
– Routing cost over each trip: predetermined fixed cost + a variable cost
depending on total distance.
– Inventory cost at each retailer: cycle inventory + safety stock.
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Problem Formulation
(6)
 v  nN fn Zvn
number of trips for route v in one year
d v   i , jR dij X ijv distance of route v
0
ltr 

1
vV
 v Rvi


lead time for retailer r. Lead time is a function
dR
vV v vi of routing route frequency (first component)
and route distance (second component)
s
Then the objective function is to Minimize:
  pP 0.5 pi

  pP z pi  vV ltr 
vV  a  cdv   v  iR hpi 

R

  vV vi v

Routing cost and inventory cost. Inventory at each retailer includes both cycle
inventory to meet foreseeable demand and safety stock to overcome uncertain
demand.
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Problem Formulation


vV
Rvi  1
pP ,iR
 pi Rvi
i  R
Single assignment
q
Vehicle capacity
v  V
v
d v  D v  V
Distance capacity
M sv  M tv  | R |  X stv  | R | 1

sR0
X stv   sR X tsv
t  R0 , v  V
0
 R  1  iR X 0iv   s ,tR
0


tR0
X itv  Rvi
nN
Z vn  1
X stv
Subtour elimination
Flow conservation
v  V
Every route starts from
DC
Variable connection
i  R, v  V
v  V
Rvi , X stv , Z vn  {0,1}
M iv  0
s, t  R, v  V
Route frequency
i  R, s, t  R0 , v V , n  N
i  I , v  V
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Modified Sweep (MS)
•
Evidence indicates that the sweep method for routing vehicles is computationally
efficient and produces an average gap from optimality of about 10 percent.
1
2
12
3
11
4
10
DC
6
9
5
7
8
•
Modify the simple sweep method by considering specific characteristics in this
problem
– Optimize routing tour after inserting a new retailer point.
– Optimize routing frequency within one route.
– Start from each retailer and sweep both clockwise and counterclockwise.
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Tabu Search-Simulated Annealing (TS-SA)
•
Distance between two routes:
The smallest possible distance between two retailers in two distinct routes. Let
Sk be the set of retailers included in route k; Dij be the distance between retailer
i and j, and DRmn be the distance between route m and n, then:
DRmn  argmin{Dij } i  Sm , j  Sn
•
Adjacent route:
Two routes are called adjacent if the distance between these two routes is
smallest compared to other routes (or within some predetermined value).
Move 1: Exchange their delivery order within one route.
Move 2: Exchange two retailers from two adjacent routes.
Move 3: Insert one retailer to an adjacent route.
Move 4: Open a new individual route for one retailer.
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Integrated Local Search Method (ILS)
•
Methodology
Generate an initial solution where each retailer is serviced by one individual
tour, and then try to merge retailers into one route.
•
Natural frequency
The optimal routing frequency for each retailer under an individual tour.
ILS1: Fixed vehicle cost + variable cost from DC to the retailer.
ILS2: Fixed cost + Variable cost (twice the distance from the DC)
ILS3: Fixed cost + Variable cost (Distance limitation)] / Average number of
retailers in one route
Whether to merge two retailers depends on two factors: the distance
between these two retailers and similarity in natural frequency.
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Integrated Local Search Method (ILS)
Procedures:
1. Calculate natural frequency for each retailer.
2. Divide all retailers into different groups. In this research, four groups will be
generated with routing frequency to be 350, 175, 50, and 25, respectively. G1,
G2, G3 and G4.
1
2
12
3
11
4
10
DC
6
9
5
7
8
Natrual Frequency
Set of retailers with the natrual freqency falling in (0, 25]
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Integrated Local Search Method (ILS)
3. Use embedded modified sweep method to merge retailers in group G1.
4. Try to insert other retailers in other groups (in the order of G2, G3 and G4) in
current routes.
1
13
Route 1
2
3
Route 2
DC
DC
5. Repeat the same process of step 3 and 4 for retailers in group G2, G3 and G4
respectively.
(6). Improvement step: Tabu search. This step is not necessary.
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Integrated Local Search V.S. Modified Sweep
r2
60 mile
r3
60 mile
100 mile
r4
r1
ILS method provides a
better solution
200 mile
100 mile
r2
100 mile
A
B
r3
200 mile
100 mile
r1
200 mile
200 mile
150 mile
MS method provides a
better solution
200 mile
r4
200 mile
150 mile
A
200 mile
150 mile
B
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Hybrid Genetic Algorithm Method (HGA)
•
•
•
Fixed partition policy (FPP)
The retailers are partitioned into disjoint
and collectively exhaustive sets. Each set
of retailers is served independently of the
others and at its optimal replenishment
rate.
Generate fixed partitions
For each fixed partition
Find optimal delivery tour (TSP)
Decide optimal delivery frequency
NO
Calculate total cost of each fixed partition
Use a genetic algorithm (GA) to
generate/update a fixed partition for all
retailers.
A TSP is solved within each partition and
optimal delivery frequency is selected
accordingly.
Update the best fixed partition
Is the termination
criterion satisfied
YES
STOP
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Bounds
•
Objective Values
The major benefit of routing comes from reduction in delivery cost. In an ideal case:
–
–
IRCLB
Delivery distance to one retailer is 1 + 1/(N+1) times the distance between nearest neighbors.
Smallest total number of trips required is total demand over all retailers dividing by a truck
capacity.


1 

1




 dr
  rR , pP  pr 
 pr

1 
1
N

1

p

P



   rR c 1 
 a
d r  r  hr 
  pP z  pr


 
 2 r
C
N 1
r
p











 
Alternatively, consider delivery distance to each retailer as D/n, where D is the distance limit
and n is the average number of retailers in one route.
 a  cD 
  pP  pr
1 D 

IRCe   rR 
  pP z  pr

  r  hr 
2 r
 r np  
 n 



Any feasible solution is an upper bound, a simple solution is using all direct-shipping.
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Parameter Settings
Name
Notation
Value
Service level
zα
1.96
Vehicle capacity
Distance limit
Vehicle speed
Fixed cost
Variable routing cost
Available
frequency/year
Location of DC
Number of retailers
Locations of retailers
C
D
s
a
cd
fn
150
500 miles
500 miles/day
$ 5/truck
c = $ 0.1mile d = distance (miles)
{25, 50, 175, 350}
Demand mean/year
Demand deviation/year
Holding cost
0
N
(x, y)
r
r
hr
Remark
97.50%
(0, 0)
{20, 50, 100, 150, 200}
[-100, 100]
10% Low: [50, 150] 80% Medium: [500, 2000]
10% High: [10000, 25000]
Low: [1, 5] High: [10, 50]
1year = 350 days
Uniform Distribution
Uniform Distribution
Uniform Distribution
Low: $ 10/unit year Medium: $ 50/unit year
High: $ 100/unit year
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Results and Analysis
CPU Seconds
Cost in $1,000
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Results and Analysis
Savings vs. Direct (Individual) Delivery
hr
High
High
Medium
Medium
Low
Low
σr
High
Low
High
Low
High
Low
N = 20
37.4
42.1
29.7
29.9
29.6
25.8
N = 50
44.2
44.6
40.5
32.3
33.7
32.2
N = 100
45.5
48.1
42.4
38.6
37.5
35.9
N = 150
47.9
51.4
42.1
38.5
37.2
37.5
N = 200
48.4
51.3
48.6
42.6
37.6
37.2
Average
44.7
47.5
40.7
36.4
35.1
33.7
• All heuristics except HGA work well in terms of objective values.
• Using routing strategy can reduce total cost by 25.8% - 51.4%.
• When the holding cost and demand variance decrease, the benefits of routing
strategy also decreases.
• Routing strategy will have more benefits if the demand or optimal order size of
each retailer is small compared to vehicle capacity.
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Results and Analysis
•
Among all heuristics, Modified Sweep method performs the best and HGA method
is the worst.
– Modified sweep method already captures many important aspect of this routing
problem.
– With capacity and distance constraints, there is a high probability that a child
from crossover and mutation is infeasible, especially in large instances.
•
ILS works very fast in terms of CPU time, but its objective values are much higher
than MS.
If joint with Tabu search, ILS-TS generates better results than MS in large
scenarios, but CPU time increases because of Tabu search step.
•
Use MS method for IRP in this research stage, and we can also
use TS to further improve results from MS method if necessary.
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Integrated Problem’s Results and
Analysis
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Integrated Problem’s Results and Analysis
Two-phase solution approach:
•
•
•
•
Phase I: Multi-product FLP with
Approximated IRC Function
DC Locations and retailer/plant
warehouse assignments
– TS-SA method
Phase II: Inventory Routing Problem
Routing tours and frequencies: solve the
routing problem for each open DC and
retailers assigned to it.
– MS method ( + TS)
Original network design problem
Phase I: FLP with approximate IRC
DC locations and retailer
assignments
Each opened
DC
Phase II: IRP, route assignments
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Results and Analysis
• Heuristics work well in terms of objective values compared to the
original greedy solution. The original greedy solution’s value is
reduced by 25.3% on average.
• The IRC gap in the table is calculated as |Real IRC/Approximated
IRC - 1|. The average value for this gap is 5.6%.
• Reasons for usingrji   ji   ji  / 2
Using average cost
5.6%
Using only routing cost
14.4%
Using direct shipping
cost
32.8%
rji  w ji  (1  w)  ji
D C
w f  , 
d 
Location
density
Demand
density
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Consolidation Facility Location and
Demand Allocation Model
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Consolidation Facility Location and Demand Allocation Model
•
Each production facility ships its product
directly to each opened DC.
•
Two shipment methods: direct shipment from
facility and indirect shipment through a DC.
•
Product sets: consolidation is allowed for
shipping products in the same product set.
•
Each facility provides one specific product
•
Single route for each product
•
Retailers hold safety stock only if the leadtime of replenishing one order is above a
specific threshold value
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Problem formulation
Fixed cost
  Q2ik h4ik
f kj wkj    i
  z h4ik

2
k
j
k
i
 k


r  ir2 lt2ik xikr 


  Q3skr hrs   Q1ir h5ir

i
  k s
  z h5ir    ir t3kr xikr lt3kr
 r
2
r
i
 k

Inventory costs at
DCs
Inventory costs at
retailers

  ir t1ir xir lt1ir  


Dir xikr

 Dir xir


 Q1ir 
 Q2ik 
    A1ir  a1ir 
 bl1ir Q1ir 
   A2ik  a2ik 
 bl2ik Q2ik  r



 Q

 Q
r
i 
k
i 
2 ik
 C1ir 
 C2ik 
 1ir


Dir xikr 

s




Q

   A3kr  a3kr  3kr   bl3kr Q3skr  iS s
 Q


C3kr 
k
r
s
Transportation cost
3 kr





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Problem formulation
Q2ik   Mwkj
i, k
Q3skr   Mwkj
s, k , r
Open DCs before
assignment
j
j
Q3skr   Q3ikr
s, k , r
w
kj
1
k
x
ikr
 xir  1
i, r
Total shipping quantity of
one product set
iS
DC’s capacity level
j
Shipment mode
k
Q2ik
i 2  i

r
2
ir
lt2ik xikr  0.8U kj wkj
k
DC’s capacity constraint
j
Q1ir , Q2ik , Q3skr  0 i, s, k , r
xir , xikr  0,1
i, k , r
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Genetic Algorithms
xir
i
k
xikr
i1
k1
r1
i2
k2
r2
r
Shipment directions
Feasible solution
example
Chromosome representation
From i to
r
1
0
0
1
Product 1 (P1)
From i to
From k to r
k
1
0
0
1
0
2
3
4
5
6
0
7
Product 2 (P2)
From i to
From i to r
From k to r
k
0
0
1
0
1
1
0
0
8
9
10
11
12
13
14
15
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Greedy Construction Heuristic
•
Build the solution step by step using a “cascade” method.
Procedures:
• Set t = 0.
• To build a table with “I R” rows and “K+1” columns and evaluate the feasibility. If
feasible, then calculate the objective function for each product-mode-retailer
combination OFikr(t) comparing the K+1 possibilities of shipment (directly by
plant or by K DCs). Otherwise put the OFikr(t) equal to a big integer called M.
• Comparing the value of OFikr(t) for each row to select the minimum and the
second smallest for each row respectively called Minir= mink{OFikr(t)}, SecMinir =
mink {OFikr(t)/Minir}.
• Calculate Δir as the difference between Minir and SecMinir (potential regret).
• Select the maxir{ Δir } and in correspondence to the column, fix the solution for
the relative product/retailer couple. Set t = t +1.
• Repeat the steps 2-5 for “I R” iterations.
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Parameter Settings
•
•
•
•
•
•
•
•
Two different product sets in the tests.
Production facilities and retailers are chosen as major cities in the United
States. Potential DCs can be located at the locations of retailers.
Each DC has three possible sizes: small, medium and large.
Fixed location cost: home value, Daskin (1995).
Capacity: potential service amount.
Demands of products: normally distributed. The mean is proportional to the
population around that retailer. The variance of demand is calculated using
coefficient of variation times mean demand.
Lead time: distance.
Shipping cost: fixed cost + variable costs (distance, shipping quantity).
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Scenarios construction
Scenario
No.
Plants
No.
DCs
No.
Retailers
Function to define
the batch size
Length of
Chromosome
1
2
3
4
5
6
7
8
2
2
5
5
2
2
5
5
10
10
10
10
10
10
10
10
10
10
10
10
49
49
49
49
Floor
Ceiling
Floor
Ceiling
Floor
Ceiling
Floor
Ceiling
140
140
600
600
608
608
2745
2745
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Results and Analysis
Heuristic
(Max {Delta = SecMin – Min})
Genetic
No.
Time
DC
(sec)
s
Time
(sec)
No.
Iter.
Obj.
Value
1
98
422
3.057E+8
4
2
123
329
3.055E+8
3
1435
9238
4
1203
5
6
All
Directshipping
Heuristic (Min {Min})
No.
Iter.
Obj.
Value
No. Time No.
DCs (sec) Iter.
Obj.
Value
No.
DCs
Obj.
Value
1
20
3.030E+8
3
1
20
3.030E+8
3
4.211E+8
4
1
20
3.030E+8
3
1
20
3.030E+8
3
4.211E+8
1.181E+9
4
3
50
9.827E+8
6
3
50
9.846E+8
7
1.180E+9
1399
1.180E+9
4
3
50
9.791E+8
7
2
50
9.846E+8
7
1.180E+9
80
652
5.060E+7
0
10
98
5.060E+7
0
10
98
5.060E+7
0
5.060E+7
39
589
5.062E+7
0
10
98
5.062E+7
0
9
98
5.062E+7
0
5.062E+7
7
17771 23760 1.743E+8
1
49
245
1.545E+8
0
25
245
1.545E+8
1
1.545E+8
8
20486 25900 1.515E+8
1
49
245
1.545E+8
1
24
245
1.545E+8
1
1.545E+8
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Results and Analysis
• The heuristic proved computationally efficient and provided the best
solution in all but one case (scenario 8).
• The “delta” form of the heuristic (making the selection based on
difference between the best and second best options) outperformed
the “min” form in two cases and the “min” form performed best in one
case.
• Both heuristics perform at least as well as direct shipments.
• GA found the unique best feasible solution in the last case but requires
significantly longer computation time.
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Conclusion
•
•
•
•
•
•
•
Proposed an innovative framework for a multi-product integrated supply
chain network design problem.
Derived and evaluated the effectiveness of a two-phase solution
methodology.
Transshipment is allowed between DCs, and routing delivery strategy is
considered.
Heuristics are generated in each phase to find a good solution in a
reasonable time.
Phase I: TS-SA method with an initial solution starting minimizing total IRC.
Phase II: MS method.
Only the special case of the original problem where only one PW is allowed
for each plant is discussed in detail in current research.
PASI 2013
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Future Needs
• Better
understanding of need for integrated formulations
• Better models for iterating/decomposing between levels (?)
• More realistic models – (problem driven not analysis driven)
• Integration of production and distribution decisions
• Taxonomy of actual problems by industry, logistics method
• Expansion to stochastic optimization (two-stage)
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Thanks!
Q&A
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