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Logistics Systems Design: Supply Chain Models March 18, 2003

Copyright © 1994-2003, Marc Goetschalckx, All rights reserved. 9 1

18-Mar-03 Logistics Systems Design © Marc Goetschalckx

Logistics Systems Design:Supply Chain Models

10. Facilities Design11. Computer Aided Layout12. Layout Models

7. Continuous Point Location8. Discrete Point Location9. Supply Chain Models


Supply Chain Models OverviewIntroduction

Global ModelsRobustness and Flexibility Models

Single Country (Domestic) ModelsChannel Selection Models


Current Algorithms Hierarchy: Benchmarking

Evaluate or benchmarkDigital simulation or spreadsheet

Distribution channel selectionCurrent facilities and capacitiesSelect best channel and inventory levelsSpreadsheet or custom programming


Current Algorithms Hierarchy: Network Optimization

Network optimizationCurrent facilities and capacitiesOptimize the product flowLinear programming solver




Current Algorithms Hierarchy:Location-Allocation

Moving facilitiesCurrent facility statusNo site-dependent costsCosts proportional to flowLimited location resolutionAlternative generating algorithm

Approximate algorithmsNon-linear optimization, specialized heuristics


Current Algorithms Hierarchy:Supply Chain Models

Supply chainEverything is allowed to changeAlternative selectingSite dependent costsMixed integer programming solver


Supply Chain Modeling Characteristics

Many different approaches existPhilosophically differentTradeoff between effort and accuracy

Tradeoff between accuracy and simplicity

DataComputational solution effortSolution validation and interpretation

Validation of computer results a must18-Mar-03 Logistics Systems Design © Marc Goetschalckx







Distribution Channel Selection

Given supply chain configurationNo fixed facility costs

For each commodityDetermine transportation channelDetermine cycle and safety inventory

18-Mar-03 © Marc Goetschalckx

Transportation Mode Selection Computations

( )( ) ( )

( )2

2

2

p p p

mp p mp

mp p p m p mp

imp mp p

imp mp p mp mp

mp mp mp LT p

mp mp p mp mp

PC D pcTC D tcPIC D v TT HCF D pic

OCIC TB v HCF

DCIC TB v tc pic HCF

SI k LT CV Var d

SIC SI v tc pic HCF

= ⋅

= ⋅

= ⋅ ⋅ ⋅ = ⋅

= ⋅ ⋅

= ⋅ + + ⋅

= +

= ⋅ + + ⋅


Transportation Mode Selection Total Cost Computations

jmp mp mp p

p p

mp mp mp

imp jmp jmp

mp jmp

mp p mp mp

FSC TB SI sc

TIC PCTVC TC PIC

OCIC DCIC SICTFC FSCTAC TIC TVC TFC

= + ⋅ =

= + +

+ +

=

= + +


Channel Selection Calculations:Ballou Example

Annual Demand 700000Annual Holding Cost Rate 0.3Unit Production Cost $30.00Unit Annualized Warehouse Cost $0.00

Rail Piggyback Truck AirUnit Transportation Cost ($) 0.1 0.15 0.2 1.4Channel Transit Time (days) 21 14 5 2Transportation Batch Size 70000 35000 35000 17500

Rail Piggyback Truck AirProduction Cost $21,000,000 $21,000,000 $21,000,000 $21,000,000Total Invariant Costs $21,000,000 $21,000,000 $21,000,000 $21,000,000Transportation Costs $70,000 $105,000 $140,000 $980,000In-Transit Inventory $362,466 $241,644 $86,301 $34,521Order Fequency per Year 10 20 20 40Order Cycle Length in Years 0.1 0.05 0.05 0.025Plant Max Cycle Inventory 70,000 35,000 35,000 17,500Plant Cycle Inventory Costs $315,000 $157,500 $157,500 $78,750Plant Safety Inventory 65,000 29,000 24,500 11,250Plant Safety Inventory Costs $585,000 $261,000 $220,500 $101,250Unit Value at DC $30.62 $30.50 $30.32 $31.45DC Max Cycle Inventory 70,000 35,000 35,000 17,500DC Cycle Inventory Costs $321,487 $160,100 $159,197 $82,554DC Safety Inventory 65,000 29,000 24,500 11,250DC Safety Inventory Costs $597,047 $265,308 $222,876 $106,141DC Maximum Inventory 135,000 64,000 59,500 28,750Total Marginal Costs $2,251,000 $1,190,552 $986,375 $1,383,216Annualized Warehouse Costs $0 $0 $0 $0Total Variant Costs $2,251,000 $1,190,552 $986,375 $1,383,216Total Cost $23,251,000 $22,190,552 $21,986,375 $22,383,216








Single Country Supply Chains: Models and Solution Algorithms

Kuehn and HamburgerDrop ñ add - swap heuristic

Geoffrion and GravesBendersí decomposition

Goetschalckx and WeiDisaggregated MIP formulation withbranch and bound with LP relaxation


Kuehn and Hamburger Characteristics

Multicommodity (as an extension)Zero echelonUncapacitated depotsDepot handling cost (as an extension)DeterministicSingle period


Kuehn and Hamburger Characteristics Continued

Equivalent path or arc formulation (one echelon)No customer single sourcingìWeakî formulation




Kuehn and Hamburger Model

1 1

1

1

min

. . 1

0

{0,1}, 0

N M

j j ij ijj i

N

ijj

M

ij ji

j ij

z f y c x

s t x i

x My j

y x

= =

=

=

= +

= ∀

− ≤ ∀

∈ ≥

∑ ∑

∑

∑


Kuehn and Hamburger Formulation Characteristics

Fixed facility cost (f) and variable transportation cost (c)Flow (x) modeled as fraction of total customer demandUncapacitatedTwo type of constraints

Demand satisfactionLinkage or consistency


Site Relative Costui = current best cost for servicing customer iρj = relative cost for opening warehouse j

{ }1

( ) min 0,M

j j ij ii

f c uρ=

= + −∑U


Drop-Add-Swap HeuristicDropï Start with all facilities openï Close the one with the largest relative

site costï Until no further decrease in costs

Addï Start with all facilities closedï Open the one with the most negative

relative site costï Until no further decrease in costs




Drop-Add-Swap Heuristic

Swapï Start with dropï Swap (open and close) the two facilities

with the most extreme relative site costï Until not further cost reduction

Kuehn and Hamburger, (1963), ì A Heuristic Program for Locating Warehouses,î Management Science, Vol. 9, pp. 643-666.


Supply Chain Example:Schematic

M1

M2

M3

C1

C2

C3

S1

S2

S3

W1

W2

W3

W4


Supply Chain Example:Transportation Data


Supply Chain Example:Capacity and Facility Data




Supply Chain Example: Model Purchasing Flows


Supply Chain Example: Model Distribution Flows


Supply Chain Example: Model Capacity and Facility Costs


Supply Chain Example: Model Excel Solver




Supply Chain Example: Solution Flows (Purchasing)


Supply Chain Example: Solution Flows (Distribution)


Supply Chain Example: Solution Costs


Arc Based FormulationsFrom an origin facility to a destination facilityAdvantages

Fewer variablesSimpler modelScales better to multiple echelons

DisadvantageNumerous conservation of flow constraints




Arc Based Conservation of Flow Constraints

10 manufacturing plants20 distribution centers200 customers40 products20*40=800 flow conservation constraints


Path Based Formulations

Flow from original supplier to final customerAdvantages

Allows path related constraints (time to market, system safety inventory)

DisadvantagesNumber of variables explodes with echelons


Path Based FormulationNumber of Variables

10 manufacturing plants20 distribution centers200 customers40 products10*20*200*40 =1.6 million variables


Geoffrion & Graves Characteristics

MulticommoditySingle echelonCapacitated depots (lower and upper bound)Depot handling costDeterministicSingle Period




Geoffrion & Graves Characteristics Continued

Path formulationCustomer single sourcingìWeakî formulationAdditional linear constraintsin z and y


Geoffrion and Graves Model

min

. .

1

ijkp ijkp j j j kp jkijkp j kp

ijkp ipjk

ijkp kp jki

jkj

j j kp jk j jpk

c x f z h r y

s t x s ip

x r y jk

y k

TL z r y TU z j

+ +

≤ ∀

= ∀

= ∀

≤ ≤ ∀

∑ ∑ ∑

∑

∑

∑

∑


Benders Primal DecompositionFix binary variables z and yï Solve independent commodity network

flow problemsï Determine total transportation cost

Add total transportation cost as a cut to binary master problemSolve binary master problemIterate


Benders Decomposition Flow Chart

Primal BinaryMaster Problem

PrimalNetwork FlowSubproblems

Transportation CostAdditional Constraint

Binary Variablesz and y




Primal Network Subproblemmin

. . [ ]

[ ]

c x

s t x s v

x r y u

x

ijkp ijkpijkp

ijkpjk

ip ip

ijkpi

kp jk jkp

ijkp

∑

∑

∑

≤

=

≥ 0

Note that y are parameters, not variables18-Mar-03 Logistics Systems Design © Marc Goetschalckx

Dual Network Subproblem

max

. .

,

y v s u r y

s t u v c ijkp

v u unrestricted

ip ip jkp kp jkjkpip

jkp ip ijkp

ip jkp

0

0

= − +

− ≤ ∀

≥

∑∑

Note that y are parameters, not variables


Extreme Point Formulation and Constraint

y v s u r y eipe

ip jkpe

kp jkjkpip

0 ≥ − + ∀∑∑

max,v u E

ipe

ip jkpe

kp jkjkpipip

ejkpe

y v s u r y∈

= − + ∑∑0


Primal Binary Master Problem

min

. .

..

, { , },

f z h r y y

s t y k

TL z r y TU z j

y v s u r y e E

z y y

j j j kp jkkpj

jkj

j j kp jkpk

j j

ipe

ip jkpe

kp jkjkpip

+FHG

IKJ

+

= ∀

≤ ≤ ∀

≥ − + =

∈ ≥

∑∑

∑

∑

∑∑

0

0

0

1

1

0 1 0




Benders Decomposition Method References

Geoffrion and Graves, (1974). "Multicommodity Distribution System Design by Benders Decomposition." Management Science, Vol. 20, No. 5, pp. 822-844.Lasdon, (1970). Optimization Theory for Large Systems. MacMillan Publishing Co., New York, New York.Van Roy, (1986). ìA Cross Decomposition Algorithm for Capacitated Facility Location.î Operations Research, Vol. 34, No. 1, pp. 145-163.


Supply Chain Ballou Example:Schematic

M1

M2

C1

C2

C3

W1

W2


Supply Chain Ballou Example:Transportation Data


Supply Chain Ballou Example:Capacity Data




Supply Chain Ballou Example:Arc Transportation Model


Supply Chain Ballou Example:Arc Capacity Model


Supply Chain Ballou Example: Arc Solver


Supply Chain Ballou Example:Arc Transportation Solution




Supply Chain Ballou Example:Arc Capacity Solution


Supply Chain Ballou Example:Path Transportation Model


Supply Chain Ballou Example:Path Capacity Model


Supply Chain Ballou Example: Path Solver




Supply Chain Ballou Example:Path Transportation Solution


Supply Chain Ballou Example:Path Capacity Solution


Logistics Modeling TrendsComputer software trends

More capable MIP solversModeling languagesERP systems (with some optimization)

Acceptance of integrated supply chain view

More comprehensive models (cradle to grave)More realistic models (inventory)


Logistics Modeling Trends:Conclusions

Models growing in complexity and realismModels developed by supply chain ownersSolution techniques become more generic and ìshrink wrappedî




SMILE CharacteristicsMulticommodityMulti-echelonCapacitated facilitiesCapacitated channelsAll costsDeterministicSingle periodDomestic (single country)


SMILE Characteristics (2)

Arc formulationSafety inventory proportional to demand (user determined)Production cost and capacitiesWeight and volume capacitiesDepot and product single sourcingSmart (tight) formulationsCPLEX MIP module


Cycle Inventory versus Replenishment Cycle Time

Inve

ntor

y

Time

AverageInventory

AverageInventory

2

2ijmp ijm

d CTCI

x R

⋅=

⋅=


Number of CarriersWeight capacity

Wt CarrierWtCapp ijmp ijm ijmp P

x w∈

≤∑

Vol CarrierVolCapp ijmp ijm ijmp P

x w∈

≤∑Volume capacity

wijm = # carriers per time periodxijmp=flow of product p per time period




Number of Carrier Constraints

25 distribution centers400 customers3 products25*400*3*2 = 60,000 constraints


Transportation CostsCarrier costMaterial flow cost

( , )

CarrierCost

TranUnitCostijm ijm

ijmp ijmpi B D j C D m M i jp P

w

x∈ ∪ ∈ ∪ ∈

∈

+

∑ ∑ ∑ ∑


Safety Inventory VersusLead Time

2d LTSI k LT Var d Var= ⋅ ⋅ + ⋅

( )2

dd

d d

VarCV

dVar CV d

=

= ⋅

2d LTSI k LT CV Var d= ⋅ ⋅ + ⋅


Safety Stock Inventory Factor

Customer service measured by probability of a stock-outSafety inventory commonly equals time length multiplied by demand (linear safety inventory policies)




Demand During Lead Time and Stock-out Probability

Prob

abilit

y

Demand

d.LT

CD=95 %

kσ


Warehouse Inventory Cost

( , )

Rr Value SSFactor

2ijm

p jp ijmpi B D j D m M i j p P

x∈ ∪ ∈ ∈ ∈

+

∑ ∑ ∑ ∑

Time

Inventory

Q

SS

0

t t+R


Machine Resources Schematic

C u s t o m e r s

Finishing Facility

P l a n t

M i l l

W a r e h o u s e

Facility

Presses

Facility

Gluers

Plant

F i n i s h i n g F a c i l i t i e s

Fixed Cost

Fixed & Var. Cost

Fixed Cost

Fixed & Var. Cost


SMILE Case Studies

Hospital supply company

State of New Jersey recycling

Electronics manufacturing

HAC wholesaler

Paper packaging supplier




Cardboard Manufacturer Supply Chain Schematic

C u s t o m e r s

Finishing Facility

P l a n t

M i l l

W a r e h o u s e

Facility

Presses

Facility

Gluers

Plant


Fixed Cost

Fixed & Var. Cost

Fixed Cost

Fixed & Var. Cost


Seasonal Demand Pattern (Soft Drinks)

0

200

400

600

800

1000

1200

1400

1600

1800

jan feb mar apr may jun jul aug sep oct nov dec


Seasonal Material Flow Schematic

Customers

Warehouses

FinishingFacilities Warehouse

sProductionFacilities

Suppliers


Initial Supply Chain




Single Season Improved Supply Chain


Cardboard Packaging Manufacturer

3 seasons, 2 stage manufacturing12 products, 714 customers87 production lines3738 transportation channels48888 variables (102 integer)primal decomposition reduced run times from 2 days to 865 secondssavings $8.3 million


Seasonal Improved Supply Chain








Global Logistics Systems Models

Domestic plus exchange rates, duties, taxesObjective is worldwide after-tax profit maximizationDecisions are material flows, transportation cost allocations, and transfer prices


Tax Rates and Profit Realization34%

17%12%

40% or more


Global Transactions and Tax Rates

a x,t,p b

Country A Country B

Tax

Rat

e

Net Income Before Tax


Impact of Transfer Prices on After Tax Profits

-120000

-100000

-80000

-60000

-40000

-20000

0

20000

40000

60000

80000

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Transfer price t

Net income after tax

x = 20,000

x = 15,000

x = 12,000

x = 8,000

x = 6,000

A loses, B profits

A and B profit

A profits, B loses

A and B lose




Before and After Tax Profit Objective and Constraint

( )

( ) ( , ) ( ) ( , )

(1 ):

1 1

k k k

lp klmp kp klm p klmpl C k m T k l p P l C k m T k l p Pl k

kpklm klm kp klm

p k

max : taxrate ibtwf ibtwfsubject to

MPRICE w HANDC TRCWM W wE E

VP HTTWM CSF SHIPFREQ SSFW TTWM

E

+ −

∈ ∈ ∈ ∈ ∈ ∈

− −

− +

− + +

∑ ∑ ∑ ∑ ∑ ∑

( ) ( )

( ) ( , )

( , ) ( )

1 1 1

1

klmpl C k m T k l P

jp jkp jkm jkm p jkmpj M m T j k p P j j

fk k k

k

w

tppldc DUTY propw TRCPW W xE

FIXDC ibtwf ibtwf k WE

∈ ∈ ∈

∈ ∈ ∈

+ −

− + + −

− = − ∈

∑ ∑ ∑

∑ ∑ ∑


Computational Test Case

50 raw material suppliers8 plants, 10 distribution centers80 customers35 components, 12 finished products3.1 modes per channel10100 variables, 2900 constraints


Profit Increases for Optimal Transfer Prices

Transfer Price HeuristicsInstance Middle Tax Lower Upper

Point Rate Bound Bound1 2.4 0.2 0.8 4.12 23.2 12.1 17.1 29.23 22.6 30.2 39.9 16.24 45.6 65.0 95.2 32.1








Design of Robust and Flexible Supply Chains

Change in the mission and data is inevitable, but only techniques are sensitivity and scenario analysisNo scientific analysis or design methodology for large SC problemsNeeded measures of

Flexibility (configuration feasibility)Robustness (quality of objective)


Research Review

Extensive literature on deterministic or scenario-based supply chain designFlexibility definitions in manufacturing research (FMS) appear not applicableStochastic optimization for small problemsSome stochastic optimization for exchange rates in global systems


Robustness and Flexibility

Relative robustness, Kouvelis (1997)

Flexibility, Beamon (1998)Unused capacity in a configuration

( ) * *

* *

( )max

( )s R s s

s Ss s

z x z xz x∈

−


Hierarchical Stochastic Design Algorithm

Select a limited number of feasible supply chain configurationsFor each configuration

Sample parameters from distributionsSolve linear network flow problemsCompute expected value and variance

Select ìbestî configurationWeighted objective or efficiency frontier




Hierarchical Two-stage Formulation

. .

{0,1}, 0

Min cx dys t Ex Fy h

Hx gx y

++ ≤≤

∈ ≥

[ ]( , ). .

{0,1}

Min cx E Q xs t Hx g

x

ξ+≤

∈

( , ). .

0

Q x Min dys t Fy h Exy

ξ =≤ −

≥


Second Stage Profit Distributions (Medium Example)

4000

000

4250

000

4500

000

4750

000

5000

000

5250

000

5500

000

5750

000

6000

000

Mor

e

Best Std.Dev. (59989)Avg. Param. Value (80989)

Best mean (144989)

01020304050607080


Multicriteria Formulation

[ ] [ ]( , ) ( , ). .

{0,1}

Min cx E Q x SD Q xs t Hx g

x

ξ α ξ+ + ⋅≤

∈

But standard Deviation (SD) is not convex


Multi-objective Criteria and Efficiency Frontier (Medium)

Efficiency Frontier

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

0 10000 20000 30000 40000 50000

10011110111010010010100011011010101101110110001010010010111001101010110111111100110101100111110




Approximations

[ ] ( )1

1( , ) ,N

i

iE Q x Q x

Nξ ξ

=≈ ∑

[ ] [ ]( , ) ( , )SD Q x Var Q xξ ξ=

[ ] ( ) ( )2

2

1 1

1 1( , ) , ,N N

i i

i iVar Q x Q x Q x

N Nξ ξ ξ

= =

≈ −

∑ ∑


Two Stage Supply ChainWith Machine Resources

C u s t o m e r s

Finishing Facility

P l a n t

M i l l

W a r e h o u s e

Facility

Presses

Facility

Gluers

Plant


Fixed Cost

Fixed & Var. Cost

Fixed Cost

Fixed & Var. Cost


Cardboard Packaging Manufacturer

2 stage manufacturing + warehouse13 products, 238 customers8 plants, 9 finishing plants28+93 production lines (machines)3738 transportation channels20912 continuous flow variables140 binary (major) site and (minor) machine variables


Domestic Case Formulation Characteristics

Problem Statistics N=1 N=20 N=40 N=60 Constraints 7,822 156,440 312,880 469,320 - Inequality constraints 3,498 69,960 139,920 209,880 - Equality constraints 4,324 86,480 172,960 259,440 Variables 21,052 418,380 836,620 1,254,860 - Continuous variables 20,912 418,240 836,480 1,254,720 - Integer (binary) variables 140 140 140 140

N = Number of scenarios in master problem




ParametersCapacities, supplies, transportation, demand, and costs all log-normally distributed400 Mhz Pentium III, CPLEX 7.041 seconds for single scenariomean-value deterministic casetotal cost = 116.115 million


Acceleration Techniques Performance: Quality (Gap)

Original 1 2 3 4 5 1st Gap > 100% 31% > 100% 60% > 100% > 100% 10th Gap 60% 8% 40% 5% 60% 9%

Original 1+2 1+3 1+4 1+5 1+2+3 1st Gap > 100% 31% 31% 31% 31% 31% 10th Gap 60% 0.7% 0.1% 0.08% 0.5% 0.2%

Original 2+3 1+3+4 1+2+3+

4 1+2+3+5 All

1st Gap > 100% 60% 31% 31% 31% 31% 10th Gap 60% 3% 0.01% 0.01% 0.06% 0.01%


Acceleration Techniques Performance: Efficiency (Run Time)

Original 1 2 3 4 5 Time > 4,000 > 4,000 > 4,000 > 4,000 > 4,000 > 4,000 Iteration > 30 > 30 > 30 > 30 > 30 > 30

Original 1+2 1+3 1+4 1+5 1+2+3 Time > 4,000 > 4,000 3,860 2,180 > 4,000 3,600 Iteration > 30 > 30 26 12 > 30 23

Original 2+3 1+3+4 1+2+3+4 1+2+3+5 All Time > 4,000 > 4,000 1,500 1,380 3,050 1,890 Iteration > 30 > 30 8 7 19 7

Note: The MIP Optimal using CPLEX needs 2,700 seconds

N=20


Comparison of Run Times for Solution Algorithms

0

10000

20000

30000

40000

50000

60000

70000

80000

90000

5 10 20 30 40 50 60

NCP

U T

ime

Regular B&B (Cplex) Improved primal Original primal




Stochastic Solutions Robustness for Domestic Case

Configuration MPP SS-1 SS-2 SS-3 Average obj. value (in million $) 116.77 111.03 111.03 111.05Max 173.30 122.57 122.08 122.11Min 99.02 100.38 100.14 100.10Range 74.28 22.19 21.93 22.01Standard Deviation 0.34 0.11 0.11 0.11Absolute Gap 5.91 0.16 0.17 0.18Relative Gap 0.066574 0.001454 0.001508 0.001619

N=20, M=20, Ní=1000


Impact of Variability

Problems Std. Deviation for customer demand

Std. deviation for all other parameters

Medium variability problem 30% 10% Low variability problem 15% 5% High variability problem 40% 20%


Robustness of StochasticSolutions for Domestic Case

106

108

110

112

114

116

118

120

Low Medium HighVariability

Avg

. tot

al c

ost (

in m

illio

n $)

MPP SS(20) SS(60)0

10

20

30

40

50

60

70

80

90

100

Low Medium High

Variability

Ran

ge (i

n m

illio

n $)

MPP SS(20) SS(60)


Configurations for Varying Variability

1 2 3 4 5 6 7 8 9Determinis tic

Stochastic (Low std. deviation)Stochastic (Medium std. deviation)

Stochastic (High std. deviation)0123456789

Num

ber o

f mac

hine

sFinishing facilities




Efficiency Graph For Many Scenarios and Replications

0.00

1.00

2.00

3.00

4.00

5.00

6.00

7.00

8.00

9.00

108.00 108.50 109.00 109.50 110.00 110.50 111.00 111.50 112.00 112.50 113.00

Avg Total Cost

Std

Dev

iatio

n To

tal C

ost

avg5932559442608655858560196600486295963089629086188660144599416325362345621686189761166613755949062657


Global Supply Chain DesignNet income after tax maximizationDemand upper bound on salesBill of material (3 levels) flow balance constraints


Global Case Problem Characteristics

Problem Statistics N=1 N=10 N=20 N=60 Constraints 1,467 14,670 29,340 88,020 - Inequality constraints 402 4,020 8,040 24,120 - Equality constraints 1,065 10,650 21,300 63,900 Variables 6,894 68,310 136,550 409,510 - Continuous variables 6,824 68,240 136,480 409,440 - Integer (binary) variables 70 70 70 70


Acceleration Techniques Performance: Quality (Gap)

Original 1 2 3 4 5 1st Gap > 100% > 100% > 100% > 100% > 100% > 100% 50th Gap 41 % 27 % 18 % 21 % 41 % 29 %

Original 1+2 1+3 1+4 1+5 1+2+3 1st Gap > 100% > 100% > 100% > 100% > 100% > 100% 50th Gap 41 % 22 % 12 % 27 % 19 % 3 %

Original 2+3 1+3+4 1+2+3+4 1+2+3+5 All 1st Gap > 100% > 100% > 100% > 100% > 100% > 100% 50th Gap 41 % 5 % 12 % 4 % < 1 % < 1 %

N=20




Acceleration Techniques for Benders Decomposition

Logistics constraintsEchelon capacity, source capacity for average demand

Cut disaggregationBy scenario

Knapsack lower bounds (max. case)Primal feasible heuristicCut strengthening


Acceleration Techniques Performance: Efficiency (Run Time)

Original 1 2 3 4 5 Time >13,000 >13,000 >13,000 >13,000 >13,000 >13,000 Iteration > 60 > 60 > 60 > 60 > 60 > 60

Original 1+2 1+3 1+4 1+5 1+2+3 Time >13,000 11,900 >13,000 >13,000 12,300 10,300 Iteration > 60 56 > 60 > 60 58 51

Original 2+3 1+3+4 1+2+3+4 1+2+3+5 All Time >13,000 >13,000 >13,000 10,300 9,800 9,800 Iteration > 60 > 60 > 60 51 45 45

Note: The MIP Optimal using CPLEX needs 730,500 seconds)

N=20


Comparison of Run Times for Solution Algorithms

0

100,000

200,000

300,000

400,000

500,000

600,000

700,000

800,000

N = 5 N = 10 N = 15 N = 20 N = 60Number of scenarios

CPU

tim

e

B&B Original Primal Improved Primal


Stochastic Solutions Robustness for the Global Case

MPP SS Average NCF 51.021 54.095Std. Deviation 0.127 0.119Max NCF 66.996 68.063Min NCF 31.355 46.531Range 35.641 21.533Absolute Gap 3.166 0.092Relative Gap 0.058425 0.001694

(in million $, except the relative gap)

N=60, M=10, Ní=1000




Robustness of StochasticSolutions for Global Case

0.000010.000020.000030.000040.000050.000060.000070.000080.0000

Low Medium High

variability of stochastic parameters

rang

e

MPP SS(N=20) SS(N=60)


Computational Experiment Conclusions: Robustness

Designing for average parameters yields a dominated configuration

Worse mean and standard deviation Difference larger for more variable data

More scenarios yield more robust solution (20 to 60 sufficient)Statistics stable after 500 samplesReport multi-criteria solutions diagram


Numerical Experiment Conclusions: Algorithms

Significant smaller times for accelerated primal decomposition

Better than B&B, original primal decompositionDual decomposition worse ??

Combination of acceleration techniques provides time reductionBill of Material makes problem much harder to solve


ConclusionsGood definitions and measures for flexibility and robustness are lackingCurrent methodology is deterministic design and sensitivity or (few) scenario analysisMany-scenario solutions are more robustOnly accelerated hierarchical design algorithm fast enough




CIMPEL Structure


Strategic Algorithms ToolkitLOCAL: heuristic location-allocationSMILE: optimal MIP, simple inventoryNETWORK: optimal material flow for given facilities

SILAS: optimal NLMIP, stochastic demand and detailed inventory


Logistics Software ReferenceAndersen Consulting, ìLogistics Software,î jointly with Council of Logistics Management, Oak Brook, IL, 1992 and yearly updates.


Supply Chain Design Challenges

Integrated models are large and complex

More tactical effects (seasonal, inventory)Multi-objective performance measures

Cost/profit, flexibility, and responsiveness tradeoffs

Strategic design as a continuous effortModels, data, algorithms




Supply Chain Design Challenges Continued

Technology transfer to logistics professionals and students

Toy cases and black-box software


Supply Chain Modeling Challenges

Multiple periods, combined with tacticalPeriodic and seasonal demandDynamic strategic systemsResponsive

GlobalTaxes and profit realizationLocal contents, duty drawback

StochasticFlexibility, robustness, risk, scenarios


From a Multicommodity Case...


...and Configurationby a Current Design Tool




To Design Toolsfor the Next Century

current algorithms hierarchy: benchmarking network ...€¦ · ydetermine transportation channel...

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