Upload
others
View
4
Download
0
Embed Size (px)
Citation preview
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
18-Mar-03 Logistics Systems Design © Marc Goetschalckx
Supply Chain Models OverviewIntroduction
Global ModelsRobustness and Flexibility Models
Single Country (Domestic) ModelsChannel Selection Models
18-Mar-03 Logistics Systems Design © Marc Goetschalckx
Current Algorithms Hierarchy: Benchmarking
Evaluate or benchmarkDigital simulation or spreadsheet
Distribution channel selectionCurrent facilities and capacitiesSelect best channel and inventory levelsSpreadsheet or custom programming
18-Mar-03 Logistics Systems Design © Marc Goetschalckx
Current Algorithms Hierarchy: Network Optimization
Network optimizationCurrent facilities and capacitiesOptimize the product flowLinear programming solver
Logistics Systems Design: Supply Chain Models March 18, 2003
Copyright © 1994-2003, Marc Goetschalckx, All rights reserved. 9 2
18-Mar-03 Logistics Systems Design © Marc Goetschalckx
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
18-Mar-03 Logistics Systems Design © Marc Goetschalckx
Current Algorithms Hierarchy:Supply Chain Models
Supply chainEverything is allowed to changeAlternative selectingSite dependent costsMixed integer programming solver
18-Mar-03 Logistics Systems Design © Marc Goetschalckx
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
Supply Chain Models OverviewIntroduction
Global ModelsRobustness and Flexibility Models
Single Country (Domestic) ModelsChannel Selection Models
Logistics Systems Design: Supply Chain Models March 18, 2003
Copyright © 1994-2003, Marc Goetschalckx, All rights reserved. 9 3
18-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
= ⋅
= ⋅
= ⋅ ⋅ ⋅ = ⋅
= ⋅ ⋅
= ⋅ + + ⋅
= +
= ⋅ + + ⋅
18-Mar-03 Logistics Systems Design © Marc Goetschalckx
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
= + ⋅ =
= + +
+ +
=
= + +
18-Mar-03 © Marc Goetschalckx
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
Logistics Systems Design: Supply Chain Models March 18, 2003
Copyright © 1994-2003, Marc Goetschalckx, All rights reserved. 9 4
18-Mar-03 Logistics Systems Design © Marc Goetschalckx
Supply Chain Models OverviewIntroduction
Global ModelsRobustness and Flexibility Models
Single Country (Domestic) ModelsChannel Selection Models
18-Mar-03 Logistics Systems Design © Marc Goetschalckx
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
18-Mar-03 Logistics Systems Design © Marc Goetschalckx
Kuehn and Hamburger Characteristics
Multicommodity (as an extension)Zero echelonUncapacitated depotsDepot handling cost (as an extension)DeterministicSingle period
18-Mar-03 Logistics Systems Design © Marc Goetschalckx
Kuehn and Hamburger Characteristics Continued
Equivalent path or arc formulation (one echelon)No customer single sourcingìWeakî formulation
Logistics Systems Design: Supply Chain Models March 18, 2003
Copyright © 1994-2003, Marc Goetschalckx, All rights reserved. 9 5
18-Mar-03 Logistics Systems Design © Marc Goetschalckx
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
= =
=
=
= +
= ∀
− ≤ ∀
∈ ≥
∑ ∑
∑
∑
18-Mar-03 Logistics Systems Design © Marc Goetschalckx
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
18-Mar-03 Logistics Systems Design © Marc Goetschalckx
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
18-Mar-03 © Marc Goetschalckx
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
Logistics Systems Design: Supply Chain Models March 18, 2003
Copyright © 1994-2003, Marc Goetschalckx, All rights reserved. 9 6
18-Mar-03 Logistics Systems Design © Marc Goetschalckx
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.
18-Mar-03 Logistics Systems Design © Marc Goetschalckx
Supply Chain Example:Schematic
M1
M2
M3
C1
C2
C3
S1
S2
S3
W1
W2
W3
W4
18-Mar-03 Logistics Systems Design © Marc Goetschalckx
Supply Chain Example:Transportation Data
18-Mar-03 Logistics Systems Design © Marc Goetschalckx
Supply Chain Example:Capacity and Facility Data
Logistics Systems Design: Supply Chain Models March 18, 2003
Copyright © 1994-2003, Marc Goetschalckx, All rights reserved. 9 7
18-Mar-03 Logistics Systems Design © Marc Goetschalckx
Supply Chain Example: Model Purchasing Flows
18-Mar-03 Logistics Systems Design © Marc Goetschalckx
Supply Chain Example: Model Distribution Flows
18-Mar-03 Logistics Systems Design © Marc Goetschalckx
Supply Chain Example: Model Capacity and Facility Costs
18-Mar-03 Logistics Systems Design © Marc Goetschalckx
Supply Chain Example: Model Excel Solver
Logistics Systems Design: Supply Chain Models March 18, 2003
Copyright © 1994-2003, Marc Goetschalckx, All rights reserved. 9 8
18-Mar-03 Logistics Systems Design © Marc Goetschalckx
Supply Chain Example: Solution Flows (Purchasing)
18-Mar-03 Logistics Systems Design © Marc Goetschalckx
Supply Chain Example: Solution Flows (Distribution)
18-Mar-03 Logistics Systems Design © Marc Goetschalckx
Supply Chain Example: Solution Costs
18-Mar-03 Logistics Systems Design © Marc Goetschalckx
Arc Based FormulationsFrom an origin facility to a destination facilityAdvantages
Fewer variablesSimpler modelScales better to multiple echelons
DisadvantageNumerous conservation of flow constraints
Logistics Systems Design: Supply Chain Models March 18, 2003
Copyright © 1994-2003, Marc Goetschalckx, All rights reserved. 9 9
18-Mar-03 Logistics Systems Design © Marc Goetschalckx
Arc Based Conservation of Flow Constraints
10 manufacturing plants20 distribution centers200 customers40 products20*40=800 flow conservation constraints
18-Mar-03 Logistics Systems Design © Marc Goetschalckx
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
18-Mar-03 Logistics Systems Design © Marc Goetschalckx
Path Based FormulationNumber of Variables
10 manufacturing plants20 distribution centers200 customers40 products10*20*200*40 =1.6 million variables
18-Mar-03 Logistics Systems Design © Marc Goetschalckx
Geoffrion & Graves Characteristics
MulticommoditySingle echelonCapacitated depots (lower and upper bound)Depot handling costDeterministicSingle Period
Logistics Systems Design: Supply Chain Models March 18, 2003
Copyright © 1994-2003, Marc Goetschalckx, All rights reserved. 9 10
18-Mar-03 Logistics Systems Design © Marc Goetschalckx
Geoffrion & Graves Characteristics Continued
Path formulationCustomer single sourcingìWeakî formulationAdditional linear constraintsin z and y
18-Mar-03 © Marc Goetschalckx
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
+ +
≤ ∀
= ∀
= ∀
≤ ≤ ∀
∑ ∑ ∑
∑
∑
∑
∑
18-Mar-03 Logistics Systems Design © Marc Goetschalckx
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
18-Mar-03 Logistics Systems Design © Marc Goetschalckx
Benders Decomposition Flow Chart
Primal BinaryMaster Problem
PrimalNetwork FlowSubproblems
Transportation CostAdditional Constraint
Binary Variablesz and y
Logistics Systems Design: Supply Chain Models March 18, 2003
Copyright © 1994-2003, Marc Goetschalckx, All rights reserved. 9 11
18-Mar-03 Logistics Systems Design © Marc Goetschalckx
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
18-Mar-03 Logistics Systems Design © Marc Goetschalckx
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
18-Mar-03 Logistics Systems Design © Marc Goetschalckx
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
Logistics Systems Design: Supply Chain Models March 18, 2003
Copyright © 1994-2003, Marc Goetschalckx, All rights reserved. 9 12
18-Mar-03 Logistics Systems Design © Marc Goetschalckx
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.
18-Mar-03 Logistics Systems Design © Marc Goetschalckx
Supply Chain Ballou Example:Schematic
M1
M2
C1
C2
C3
W1
W2
18-Mar-03 Logistics Systems Design © Marc Goetschalckx
Supply Chain Ballou Example:Transportation Data
18-Mar-03 Logistics Systems Design © Marc Goetschalckx
Supply Chain Ballou Example:Capacity Data
Logistics Systems Design: Supply Chain Models March 18, 2003
Copyright © 1994-2003, Marc Goetschalckx, All rights reserved. 9 13
18-Mar-03 Logistics Systems Design © Marc Goetschalckx
Supply Chain Ballou Example:Arc Transportation Model
18-Mar-03 Logistics Systems Design © Marc Goetschalckx
Supply Chain Ballou Example:Arc Capacity Model
18-Mar-03 Logistics Systems Design © Marc Goetschalckx
Supply Chain Ballou Example: Arc Solver
18-Mar-03 Logistics Systems Design © Marc Goetschalckx
Supply Chain Ballou Example:Arc Transportation Solution
Logistics Systems Design: Supply Chain Models March 18, 2003
Copyright © 1994-2003, Marc Goetschalckx, All rights reserved. 9 14
18-Mar-03 Logistics Systems Design © Marc Goetschalckx
Supply Chain Ballou Example:Arc Capacity Solution
18-Mar-03 Logistics Systems Design © Marc Goetschalckx
Supply Chain Ballou Example:Path Transportation Model
18-Mar-03 Logistics Systems Design © Marc Goetschalckx
Supply Chain Ballou Example:Path Capacity Model
18-Mar-03 Logistics Systems Design © Marc Goetschalckx
Supply Chain Ballou Example: Path Solver
Logistics Systems Design: Supply Chain Models March 18, 2003
Copyright © 1994-2003, Marc Goetschalckx, All rights reserved. 9 15
18-Mar-03 Logistics Systems Design © Marc Goetschalckx
Supply Chain Ballou Example:Path Transportation Solution
18-Mar-03 Logistics Systems Design © Marc Goetschalckx
Supply Chain Ballou Example:Path Capacity Solution
18-Mar-03 Logistics Systems Design © Marc Goetschalckx
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)
18-Mar-03 Logistics Systems Design © Marc Goetschalckx
Logistics Modeling Trends:Conclusions
Models growing in complexity and realismModels developed by supply chain ownersSolution techniques become more generic and ìshrink wrappedî
Logistics Systems Design: Supply Chain Models March 18, 2003
Copyright © 1994-2003, Marc Goetschalckx, All rights reserved. 9 16
18-Mar-03 Logistics Systems Design © Marc Goetschalckx
SMILE CharacteristicsMulticommodityMulti-echelonCapacitated facilitiesCapacitated channelsAll costsDeterministicSingle periodDomestic (single country)
18-Mar-03 Logistics Systems Design © Marc Goetschalckx
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
18-Mar-03 Logistics Systems Design © Marc Goetschalckx
Cycle Inventory versus Replenishment Cycle Time
Inve
ntor
y
Time
AverageInventory
AverageInventory
2
2ijmp ijm
d CTCI
x R
⋅=
⋅=
18-Mar-03 Logistics Systems Design © Marc Goetschalckx
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
Logistics Systems Design: Supply Chain Models March 18, 2003
Copyright © 1994-2003, Marc Goetschalckx, All rights reserved. 9 17
18-Mar-03 Logistics Systems Design © Marc Goetschalckx
Number of Carrier Constraints
25 distribution centers400 customers3 products25*400*3*2 = 60,000 constraints
18-Mar-03 Logistics Systems Design © Marc Goetschalckx
Transportation CostsCarrier costMaterial flow cost
( , )
CarrierCost
TranUnitCostijm ijm
ijmp ijmpi B D j C D m M i jp P
w
x∈ ∪ ∈ ∪ ∈
∈
+
∑ ∑ ∑ ∑
18-Mar-03 Logistics Systems Design © Marc Goetschalckx
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= ⋅ ⋅ + ⋅
18-Mar-03 Logistics Systems Design © Marc Goetschalckx
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)
Logistics Systems Design: Supply Chain Models March 18, 2003
Copyright © 1994-2003, Marc Goetschalckx, All rights reserved. 9 18
18-Mar-03 Logistics Systems Design © Marc Goetschalckx
Demand During Lead Time and Stock-out Probability
Prob
abilit
y
Demand
d.LT
CD=95 %
kσ
18-Mar-03 Logistics Systems Design © Marc Goetschalckx
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
18-Mar-03 Logistics Systems Design © Marc Goetschalckx
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
18-Mar-03 Logistics Systems Design © Marc Goetschalckx
SMILE Case Studies
Hospital supply company
State of New Jersey recycling
Electronics manufacturing
HAC wholesaler
Paper packaging supplier
Logistics Systems Design: Supply Chain Models March 18, 2003
Copyright © 1994-2003, Marc Goetschalckx, All rights reserved. 9 19
18-Mar-03 Logistics Systems Design © Marc Goetschalckx
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
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
18-Mar-03 Logistics Systems Design © Marc Goetschalckx
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
18-Mar-03 Logistics Systems Design © Marc Goetschalckx
Seasonal Material Flow Schematic
Customers
Warehouses
FinishingFacilities Warehouse
sProductionFacilities
Suppliers
18-Mar-03 Logistics Systems Design © Marc Goetschalckx
Initial Supply Chain
Logistics Systems Design: Supply Chain Models March 18, 2003
Copyright © 1994-2003, Marc Goetschalckx, All rights reserved. 9 20
18-Mar-03 Logistics Systems Design © Marc Goetschalckx
Single Season Improved Supply Chain
18-Mar-03 Logistics Systems Design © Marc Goetschalckx
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
18-Mar-03 Logistics Systems Design © Marc Goetschalckx
Seasonal Improved Supply Chain
18-Mar-03 Logistics Systems Design © Marc Goetschalckx
Supply Chain Models OverviewIntroduction
Global ModelsRobustness and Flexibility Models
Single Country (Domestic) ModelsChannel Selection Models
Logistics Systems Design: Supply Chain Models March 18, 2003
Copyright © 1994-2003, Marc Goetschalckx, All rights reserved. 9 21
18-Mar-03 Logistics Systems Design © Marc Goetschalckx
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
18-Mar-03 Logistics Systems Design © Marc Goetschalckx
Tax Rates and Profit Realization34%
17%12%
40% or more
18-Mar-03 Logistics Systems Design © Marc Goetschalckx
Global Transactions and Tax Rates
a x,t,p b
Country A Country B
Tax
Rat
e
Net Income Before Tax
18-Mar-03 Logistics Systems Design © Marc Goetschalckx
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
Logistics Systems Design: Supply Chain Models March 18, 2003
Copyright © 1994-2003, Marc Goetschalckx, All rights reserved. 9 22
18-Mar-03 Logistics Systems Design © Marc Goetschalckx
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
∈ ∈ ∈
∈ ∈ ∈
+ −
− + + −
− = − ∈
∑ ∑ ∑
∑ ∑ ∑
18-Mar-03 Logistics Systems Design © Marc Goetschalckx
Computational Test Case
50 raw material suppliers8 plants, 10 distribution centers80 customers35 components, 12 finished products3.1 modes per channel10100 variables, 2900 constraints
18-Mar-03 Logistics Systems Design © Marc Goetschalckx
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
18-Mar-03 Logistics Systems Design © Marc Goetschalckx
Supply Chain Models OverviewIntroduction
Global ModelsRobustness and Flexibility Models
Single Country (Domestic) ModelsChannel Selection Models
Logistics Systems Design: Supply Chain Models March 18, 2003
Copyright © 1994-2003, Marc Goetschalckx, All rights reserved. 9 23
18-Mar-03 Logistics Systems Design © Marc Goetschalckx
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)
18-Mar-03 Logistics Systems Design © Marc Goetschalckx
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
18-Mar-03 Logistics Systems Design © Marc Goetschalckx
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∈
−
18-Mar-03 Logistics Systems Design © Marc Goetschalckx
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
Logistics Systems Design: Supply Chain Models March 18, 2003
Copyright © 1994-2003, Marc Goetschalckx, All rights reserved. 9 24
18-Mar-03 Logistics Systems Design © Marc Goetschalckx
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 =≤ −
≥
18-Mar-03 Logistics Systems Design © Marc Goetschalckx
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
18-Mar-03 Logistics Systems Design © Marc Goetschalckx
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
18-Mar-03 Logistics Systems Design © Marc Goetschalckx
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
Logistics Systems Design: Supply Chain Models March 18, 2003
Copyright © 1994-2003, Marc Goetschalckx, All rights reserved. 9 25
18-Mar-03 Logistics Systems Design © Marc Goetschalckx
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
= =
≈ −
∑ ∑
18-Mar-03 Logistics Systems Design © Marc Goetschalckx
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
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
18-Mar-03 Logistics Systems Design © Marc Goetschalckx
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
18-Mar-03 Logistics Systems Design © Marc Goetschalckx
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
Logistics Systems Design: Supply Chain Models March 18, 2003
Copyright © 1994-2003, Marc Goetschalckx, All rights reserved. 9 26
18-Mar-03 Logistics Systems Design © Marc Goetschalckx
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
18-Mar-03 Logistics Systems Design © Marc Goetschalckx
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%
18-Mar-03 Logistics Systems Design © Marc Goetschalckx
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
18-Mar-03 Logistics Systems Design © Marc Goetschalckx
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
Logistics Systems Design: Supply Chain Models March 18, 2003
Copyright © 1994-2003, Marc Goetschalckx, All rights reserved. 9 27
18-Mar-03 Logistics Systems Design © Marc Goetschalckx
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
18-Mar-03 Logistics Systems Design © Marc Goetschalckx
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%
18-Mar-03 Logistics Systems Design © Marc Goetschalckx
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)
18-Mar-03 Logistics Systems Design © Marc Goetschalckx
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
Logistics Systems Design: Supply Chain Models March 18, 2003
Copyright © 1994-2003, Marc Goetschalckx, All rights reserved. 9 28
18-Mar-03 Logistics Systems Design © Marc Goetschalckx
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
18-Mar-03 Logistics Systems Design © Marc Goetschalckx
Global Supply Chain DesignNet income after tax maximizationDemand upper bound on salesBill of material (3 levels) flow balance constraints
18-Mar-03 Logistics Systems Design © Marc Goetschalckx
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
18-Mar-03 Logistics Systems Design © Marc Goetschalckx
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
Logistics Systems Design: Supply Chain Models March 18, 2003
Copyright © 1994-2003, Marc Goetschalckx, All rights reserved. 9 29
18-Mar-03 © Marc Goetschalckx
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
18-Mar-03 Logistics Systems Design © Marc Goetschalckx
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
18-Mar-03 Logistics Systems Design © Marc Goetschalckx
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
18-Mar-03 Logistics Systems Design © Marc Goetschalckx
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
Logistics Systems Design: Supply Chain Models March 18, 2003
Copyright © 1994-2003, Marc Goetschalckx, All rights reserved. 9 30
18-Mar-03 Logistics Systems Design © Marc Goetschalckx
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)
18-Mar-03 Logistics Systems Design © Marc Goetschalckx
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
18-Mar-03 Logistics Systems Design © Marc Goetschalckx
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
18-Mar-03 Logistics Systems Design © Marc Goetschalckx
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
Logistics Systems Design: Supply Chain Models March 18, 2003
Copyright © 1994-2003, Marc Goetschalckx, All rights reserved. 9 31
18-Mar-03 Logistics Systems Design © Marc Goetschalckx
CIMPEL Structure
18-Mar-03 Logistics Systems Design © Marc Goetschalckx
Strategic Algorithms ToolkitLOCAL: heuristic location-allocationSMILE: optimal MIP, simple inventoryNETWORK: optimal material flow for given facilities
SILAS: optimal NLMIP, stochastic demand and detailed inventory
18-Mar-03 Logistics Systems Design © Marc Goetschalckx
Logistics Software ReferenceAndersen Consulting, ìLogistics Software,î jointly with Council of Logistics Management, Oak Brook, IL, 1992 and yearly updates.
18-Mar-03 Logistics Systems Design © Marc Goetschalckx
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
Logistics Systems Design: Supply Chain Models March 18, 2003
Copyright © 1994-2003, Marc Goetschalckx, All rights reserved. 9 32
18-Mar-03 Logistics Systems Design © Marc Goetschalckx
Supply Chain Design Challenges Continued
Technology transfer to logistics professionals and students
Toy cases and black-box software
18-Mar-03 Logistics Systems Design © Marc Goetschalckx
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
18-Mar-03 Logistics Systems Design © Marc Goetschalckx
From a Multicommodity Case...
18-Mar-03 Logistics Systems Design © Marc Goetschalckx
...and Configurationby a Current Design Tool