Integrated Consolidation Facility Location and Inventory Routing for Supply Networks

PASI 2013Santiago, Chile

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


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, DirectorSchool of Computing, Informatics, and Decision Systems EngineeringArizona State UniversityTempe, AZ 85287-8809 [email protected]


4

Arizona State University


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


7

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


Logistics: Facility Layout to Supply Networks

• Production Control Systems• Supply Chain Design• Batch Sizing/Lot Streaming

• Queuing Networks• Material Flow & Capacity Models• Facility Layout


Introduction

• Supply Chain Management (SCM)

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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.


Supply Chain Management Decisions

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Strategic Level

Tactical Level

Operational Level

Long Term

Medium Term

Near Term

• Corporate objectives• Capacity / Facilities• Markets to operate• Location• Resources

• Aggregate planning• Resource allocation• Capacity allocation• Distribution• Inventory management

• Shop floor scheduling• Delivery scheduling• Truck routing

Facility Location Problem (FLP)

Inventory Control Problem (ICP)

Vehicle Routing Problem (VRP)


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).

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Customer service goals

Inventory Strategy• Forecasting• Inventory decisions• Purchasing and supply scheduling decisions• Storage fundamentals• Storage decisions

Transportation Strategy• Transport fundamentals• Transport decisions

Location Strategy• Location decisions• The network planning process


Our Distribution Problem

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Made Here in Volume


Distribution Problem Scenario

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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 short-term 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?


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


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

Di

Di

Di

Di

Di

1/2( )iZ Var D 1/2( )iZ Var D 1/2( )iZ Var D

1/2( )iZ N Var D

Assumes independence


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Principles to Keep in Mind2. Inventory vs. Service Level

What’s the Traditional Perspective?

Inventory

Fill Rate

100%


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


Empirical Profile: Know When Enough is Enough

0

2

4

6

8

10

12

0 10 20 30 40 50 60 70 80 90

Thro

ughp

ut

WIP

Little's Law and Chaos

Deterministic

Exponential

Empirical

Remember

L =λW

N = XT

In theory, there’s no difference between theory and practice, in practice there is. – Yogi Berra


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.


Consolidation and Transportation Centers

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DistributionCenter

Retailer

Facility

B: Only Distribution Center

ConsolidationCenter

A: Through Consolidation Center

Retailer

Facility

Facility

C: Point-to-Point Transportation

Retailer

DistributionCenter

Fixed Location Cost V.S.Transportation savingsEasy management


Multi-Product Integrated Supply Chain Network Model

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Facility DC Retailer

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Distribution RetailerConsolidationFacility

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Problem Description

– 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.

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Problem Description

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.

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Research Scope and Activities

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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.

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Facility DC

Retailer

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Retailer

DCFacility

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RetailerDCFacility

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Cost Components

FC: Annualized fixed cost of locating DCs

SC: FTL Shipping cost from plants to DCs and between DCs

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1

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1

2

3

4

Qpj

Qpj’j

' '

'' ,

' '

' ' ' '

''' , '

'

pj j pi pj jii I

pj pi pjj ij J i I

pj l pjl j j l j jll L l L

pj pjl pjl l pjl j j j jl j jl l j jll L l L

pj jpj p Pj pj j jp P j J j j

pj j j

Q Y

Q Y

q q T q q T

A a b q T A a b q T

QQSC A A

q q

, jk jkj J k KFC f O

p j, j’ i

Q = total shipped/timeq = quantity per trip (shipping mode capacity)A = cost per tripT = if use that truck size


Cost Components

SSC: Safety stock inventory holding cost at DCs

RIC: Regular inventory holding cost at DCs

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1

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Qpj

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2 2' ' ', ' ' , 'pj pj pi pjj i j j pi pj jii I j J j J j j i I

SS z lt Y lt Y

' '' , '

'2 2pj j j pj j

pj j J j jp p pj jp P

q q QRI

v v Q

Ypjj’I binary route indicator variables


Phase I: Multi-product FLP with Approximated IRC Function

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Approximated IRC Function

IRC: Annual inventory routing cost from DCs to retailersrji a computationally estimated parameter to represent the annual

IRC at retailer i if assigned to DC j

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, ji jii I j JIRC r y

max | ( ) |ji jn R i

( , ) ( ). . :

jji lml m A i

s t d D

( ), jpm pm R i p P

ji

vq

Inventory costRouting cost

Nearest insertion method

5

4

DC

1

23

6

78

9

10

11

12

1min2

ji pi jiji n pi pip Pn N

ji n n

a cf h z

n f f s


Approximated IRC Function

IRC: Annual inventory routing cost from DCs to retailersrji a computationally estimated parameter to represent the annual

IRC at retailer i if assigned to DC j

, ji jii I j JIRC r y

direct shipping method

21min 22

pi jiji ji n pi pip Pn N

ji ji

da cd f h z

s

/ 2ji ji jir


Problem Formulation

1 jij JY i I

Minimize (FC + IRC + SC + SSC + RIC)Subject to:

Single source Single path

Link variables

Maximum number of PWsSingle capacity level

Truck size selection

Throughput Capacity limit

'',1 ,pj jij j J

Y i I p P

', '

,pj ji jip P j JY MY i I j J

', ' ,pjj i pji I j J

Y MW p P j J

pj pj J

W PW p P

1 jkk K

O j J

1 ,pjll l

T p P j J

' 1 , ' , 'jj ll L

T j j J j j

', , , ' , '

pi ji pi pjj i jk jkp P i I p P i I j J j j k KY Y C O j J

' ', , , , , {0,1} , , ' , ,jk ji pj ji pj pjl jj lO Y Y W T T i I j j J p P k K


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

TSRIC: Transshipment and regular working inventory holding cost

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min2

pii I lpj pjl pjl l pjp Pl

l

qDSRIC a b q hq

' '' ' '

'

min2

pj jp P pj jlj j j jl j jl l pjp Pl

l pj jp P

Q QqTSRIC a b q hq Q

Shared transhipment loads


Nonlinear Terms

Safety stock

Working inventory

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2 2' ' ', ',pj pi pjj i j j pi pj jii I j J i I

lt Y lt Y

2 2 2 2' ' ', '

' , ''

2', '

2' ' '

pj pi pjj i j j pi pj jii I j J i Ipj j J j j

pj pj j

pj pj pi pjj ii I j J

pj j j j pi pj jii I

z lt Y z lt YSS

S S

S z lt Y

S z lt Y

'

'

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pj j

pj jp P

pj j pi pj jii I

QQ

Q Y

Recursive procedure to update safety stock parameters, Gebennini et al. (2009)

Same holding rate


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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Tabu Search-Simulated Annealing (TS-SA)


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 componentIRC 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

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Ad-Hoc Heuristics

Nothing new, sounds like variable selection in regression.


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Lower Bound: Without considering transshipment between DCs.

,

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= +

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pj j j

pj j j pj jpj pj pi pjj i j j pi pj jip P j J i I j J j J j j i I j J j j

pj jp P

pj

SC h SS RI

QQA A

q q

q q Qh z lt Y lt Y

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A

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=2

Where min

pj pjpj pj pi pjj ip P p P j J i I j J

pj

pi pj pji Ipj pj pj pi pjp P j J i I j J

pj

pj pjp P j J

pi pi Ipj l L pjl pjl l pj pj pii I j J

l

Q qh z lt Y

q

h qA h z lt W

q

W

ha b q h z lt

q

2

j lq

n =1


• 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.

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Parameter Settings

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

Low demand 10% consume 3% TD 80% consume 10% TD


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Partial Results


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Partial Results


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Results Comparison

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 1204.5

4.7

4.9

5.1

5.3

5.5

5.7

5.9

6.1

6.3

Greedy TSSA IRC-TSSA FC-TSSA LowerBound

Log of Cost (to show separation)


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Large Problems 200 Retailers, 20 DCs

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120200000

400000

600000

800000

1000000

1200000

1400000

1600000

Greedy TSSA IRC-TSSA FC-TSSA LowerBound


• 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.

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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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FC: fixed location cost; IRC: inventory routing cost;IC: regular inventory and safety stock cost at DCs; SC: total shipping cost

FCRate = Low; HCRate

= Low

FCRate = Low; HCRate

= High

FCRate = High;

HCRate = Low

FCRate = High;

HCRate = High

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

100%

21.9% 16.2% 35.9% 27.9%

57.4% 63.4% 47.1% 54.3%

6.9% 7.4% 5.4% 6.9%

13.7% 12.9% 11.7% 10.9%

SCICIRCFC


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

•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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Characteristic AlternativesTime Instant Finite InfiniteDemand Stochastic DeterministicTopology One-to-one One-to-many Many-to-manyRouting Direct Multiple ContinuousInventory Fixed Stock-out Lost sale Back-orderFleet composition Homogeneous HeterogeneousFleet size Single Multiple UnconstrainedProducts Single Multiple


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)

number of trips for route v in one yearv n vnn Nf Z

0,v ij ijvi j Rd d X

distance of route v

1 v viv Vr

v viv V

d Rlt

R s

lead time for retailer r. Lead time is a function of routing route frequency (first component) and route distance (second component)

Then the objective function is to Minimize:

Routing cost and inventory cost. Inventory at each retailer includes both cycle inventory to meet foreseeable demand and safety stock to overcome uncertain demand.

0.5 pip P

v v pi pi rv V i R p P v Vvi vv V

a cd h z ltR


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Problem Formulation

Vehicle capacity

Every route starts from DC

Flow conservation

Distance capacity

Single assignment

Subtour elimination

0 0

0

0

,

0

0 ,

1

| | | | 1 , ,

,

1

,

viv V

pi vip P i R

v

v

sv tv stv

stv tsvs R s R

iv stvi R s t R

itv vit R

vnn

R i R

Rq v V

d D v VM M R X R s t R v V

X X t R v V

R X X v V

X R i R v V

Z

0

1

, , {0,1} , , , ,0 ,

N

vi stv vn

iv

v V

R X Z i R s t R v V n NM i I v V

Variable connection

Route frequency


• Evidence indicates that the sweep method for routing vehicles is computationally efficient and produces an average gap from optimality of about 10 percent.

• 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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Modified Sweep (MS)

5

4

DC

1

23

6

78

9

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12


• 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:

• 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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Tabu Search-Simulated Annealing (TS-SA)

arg min{ } , mn ij m nDR D i S j S


• MethodologyGenerate an initial solution where each retailer is serviced by one individual tour, and then try to merge retailers into one route.

• Natural frequencyThe 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

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Integrated Local Search Method (ILS)

Whether to merge two retailers depends on two factors: the distance between these two retailers and similarity in natural frequency.


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.

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5

4

DC

1

23

6

78

9

10

11

12


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.

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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DC

32

1

13

Route 1

Route 2

DC


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Integrated Local Search V.S. Modified Sweep

BA

60 mile

100 mile

60 mile

200 mile

r4

r3r2

r1

B

200 mile100 mile

100 mile

150 mile150 mile

200 mile

200 mile

200 mile200 mile

150 mile

100 mile

A

200 mile r4

r3

r2

r1

ILS method provides a better solution

MS method provides a better solution


• 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.

• 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.

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Hybrid Genetic Algorithm Method (HGA)

Generate fixed partitions

Find optimal delivery tour (TSP)

For each fixed partition

Decide optimal delivery frequency

Calculate total cost of each fixed partition

Update the best fixed partition

Is the termination criterion satisfied

STOP

YES

NO


• Objective Values The major benefit of routing comes from reduction in delivery cost. In an ideal case:– 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.

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.

Any feasible solution is an upper bound, a simple solution is using all direct-shipping.

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Bounds

,

111 1 11

1 2

rpr prr R p P p P

LB r r r prr R p Pr r

dNIRC a c d h z

C N p

12

prp Pe r r prr R p P

r r

a cD DIRC h zn np


70

Parameter Settings

Name Notation Value Remark

Service level zα 1.96 97.50%

Vehicle capacity C 150Distance limit D 500 milesVehicle speed s 500 miles/dayFixed cost a $ 5/truckVariable routing cost cd c = $ 0.1mile d = distance (miles) Available frequency/year

fn {25, 50, 175, 350} 1year = 350 days

Location of DC 0 (0, 0)Number of retailers N {20, 50, 100, 150, 200}Locations of retailers (x, y) [-100, 100] Uniform Distribution

Demand mean/year 10% Low: [50, 150] 80% Medium: [500, 2000]10% High: [10000, 25000]

Uniform Distribution

Demand deviation/year Low: [1, 5] High: [10, 50] Uniform DistributionHolding cost hr Low: $ 10/unit year Medium: $ 50/unit year

High: $ 100/unit year

r

r


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Results and AnalysisCPU Seconds Cost in $1,000


•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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hr σr N = 20 N = 50 N = 100 N = 150 N = 200 AverageHigh High 37.4 44.2 45.5 47.9 48.4 44.7High Low 42.1 44.6 48.1 51.4 51.3 47.5Medium High 29.7 40.5 42.4 42.1 48.6 40.7Medium Low 29.9 32.3 38.6 38.5 42.6 36.4Low High 29.6 33.7 37.5 37.2 37.6 35.1Low Low 25.8 32.2 35.9 37.5 37.2 33.7

Savings vs. Direct (Individual) Delivery


• 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.

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Use MS method for IRP in this research stage, and we can also use TS to further improve results from MS method if necessary.


Integrated Problem’s Results and Analysis

74


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)

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Original network design problem

Phase I: FLP with approximate IRC

DC locations and retailer assignments

Phase II: IRP, route assignments

Each opened DC


• 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 using

76


/ 2ji ji jir

Using average cost 5.6%

Using only routing cost 14.4%

Using direct shipping cost

32.8%

(1 )

,

ji ji jir w w

D Cw fd

Demand density

Locationdensity


Consolidation Facility Location and Demand Allocation Model

77


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 lead-time of replenishing one order is above a specific threshold value

78


79

Problem formulation

2 42

4 2

3 1 5

5 3 3 1 1

11 1 1 1

1 1

2

2

ik iki

kj kj ik ir ik ikrk j k k i r

s skr r ir ir

k s iir ir kr ikr kr ir ir ir ir

r r i k

ir ir irir ir ir ir

i ir

Q hf w z h lt x

Q h Q hz h t x lt t x lt

Q D xA a bl Q

C Q

22 2 2 2

2 2

33 3 3 3

3 3

ir ikrik r

ik ik ik ikr k iir ik ik

s ir ikrskr i S

kr kr kr kr sk r s kr kr

D xQ

A a bl QC Q

D xQ

A a bl QC Q

Fixed cost

Transportation cost

Inventory costs at retailers

Inventory costs at DCs


80

Problem formulation

Shipment mode

DC’s capacity level

Total shipping quantity of one product set

Open DCs before assignment

2

3

3 3

222

1 2 3

,

, ,

, ,

1

1 ,

0.8 2

, , 0 , , ,

, 0,1

ik kjj

skr kj

j

skr ikr

i S

kjj

ikr irk

ikir ik ikr kj kj

i i r j

sir ik kr

ir ikr

Q Mw i k

Q Mw s k r

Q Q s k r

w k

x x i r

Qlt x U w k

Q Q Q i s k r

x x

, ,i k r

DC’s capacity constraint


81

Genetic Algorithms

i k r

irx

ikrx

i1 k1 r1

i2 k2 r2

Product 1 (P1) Product 2 (P2)From i to

rFrom i to

k From k to r From i to r From i to k From k to r

1 0 1 0 0 1 0 0 0 0 1 0 1 1 0 00 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Shipment directionsFeasible solution example

Chromosome representation


• 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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Greedy Construction Heuristic


• 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).

83

Parameter Settings


84

Scenarios construction

Scenario No.Plants

No.DCs

No. Retailers

Function to define the batch size

Length of Chromosome

1 2 10 10 Floor 1402 2 10 10 Ceiling 1403 5 10 10 Floor 6004 5 10 10 Ceiling 6005 2 10 49 Floor 6086 2 10 49 Ceiling 6087 5 10 49 Floor 27458 5 10 49 Ceiling 2745


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Genetic Heuristic(Max {Delta = SecMin – Min}) Heuristic (Min {Min})

All Direct-

shipping

Time (sec)

No.Iter.

Obj. Value

No. DCs

Time (sec)

No.Iter.

Obj. Value

No. DCs

Time (sec)

No.Iter.

Obj. Value

No. DCs

Obj. Value

1 98 422 3.057E+8 4 1 20 3.030E+8 3 1 20 3.030E+8 3 4.211E+8

2 123 329 3.055E+8 4 1 20 3.030E+8 3 1 20 3.030E+8 3 4.211E+8

3 1435 9238 1.181E+9 4 3 50 9.827E+8 6 3 50 9.846E+8 7 1.180E+9

4 1203 1399 1.180E+9 4 3 50 9.791E+8 7 2 50 9.846E+8 7 1.180E+9

5 80 652 5.060E+7 0 10 98 5.060E+7 0 10 98 5.060E+7 0 5.060E+7

6 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


• 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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• 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.

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Conclusion


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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)


Thanks!Q & A

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Documents

Integrated Consolidation Facility Location and Inventory Routing for Supply Networks