Supply Chain Optimization

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Supply Chain Optimization
KUBO
Mikio
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Agenda

• Definition of Supply Chain (SC) and Logistics
• Decision Levels of SC
• Classification of Inventory
• Basic Models in SC
• Logistics Network Design
• Inventory
• Production Planning
• Vehicle Routing

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Def. of SCMCouncil of SCM Professionals

• Supply chain management encompasses the planning
  and management of all activities involved in
  sourcing and procurement, conversion, and all
  logistics management activities. Importantly, it
  also includes coordination and collaboration with
  channel partners, which can be suppliers,
  intermediaries, third party service providers,
  and customers. In essence, supply chain
  management integrates supply and demand
  management within and across companies.

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Def. of LogisticsCouncil of SCM Professionals

• Logistics management is that part of supply chain
  management that plans, implements, and controls
  the efficient, effective forward and reverses
  flow and storage of goods, services and related
  information between the point of origin and the
  point of consumption in order to meet customers'
  requirements.

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Whats Supply Chain
IT(Information TechnologyLogistics
Supply Chain
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Real System, Transactional IT, Analytic IT
Analytic ITModelAlgorithm Decision Support
System
brain
Transactional ITPOS, ERP, MRP, DRPAutomatic
Information Flow
nerve
Real SystemTruck, Ship, Plant, Product, Machine,

muscle
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Levels of Decision Making
Strategic Level
A year to several years long-term decision making
Analytic IT
Tactical Level
A week to several months mid-term decision making
Operational Level
Transactional IT
Real time to several days
short-term decision making
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Models in Analytic IT
Plant
DC
Supplier
Retailer
Logistics Network Design
Strategic
Multi-period Logistics Network Design
Inventory Safety stock allocation Inventory
policy optimization
Production Lot-sizing Scheduling
Transportation Delivery Vehicle Routing
Tactical
Operational
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Models in Analytic IT
Plant
DC
Supplier
Retailer
Logistics Network Design
Strategic
Multi-period Logistics Network Design
Inventory Safety stock allocation Inventory
policy optimization
Production Lot-sizing Scheduling
Transportation Delivery Vehicle Routing
Tactical
Operational
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Models in Analytic IT
Plant
DC
Supplier
Retailer
Logistics Network Design
Strategic
Multi-period Logistics Network Design
Inventory Safety stock allocation Inventory
policy optimization
Production Lot-sizing Scheduling
Transportation Delivery Vehicle Routing
Tactical
Operational
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InventoryBlood of Supply Chain
Inventory acts as glue connecting optimization
systems
Plant
DC
Supplier
Retailer
Work-in-process
Finished goods
Raw material
Time
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Classification of Inventory

• In-transit (pipeline) inventoryInventories that
  are in-transit of productsTrade-off
  transportation cost or production
  speed-gtLogistics Network Design (LND)
• Seasonal inventoryInventories for time-varying
  (seasonal) demands Trade-off resource
  acquisition or overtime cost -gt multi-period LND
  Trade-off setup cost -gt Lot-sizing
• Cycle inventoryInventories caused by periodic
  activitiesTrade-off transportation (or
  production) fixed cost -gt LNDTrade-off ordering
  fixed cost-gt Economic Ordering Quantity (EOQ)
• Lot-size inventoryCycle inventories when the
  speed of demand is not constantTrade-off fixed
  cost -gtLot-sizing, multi-period LND
• Safety inventoryInventories for the demand
  variabilityTrade-off customer service level
  gtSafety Stock Allocation, LNDTrade-off
  backorder (stock-out) cost -gtInventory Policy
  Optimization

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In-transit (pipeline) Inventory

• Inventory that are in-transit of
  productsTrade-off transportation cost or
  transportation/production speed-gtoptimized in
  Logistics Network Design (LND)

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Seasonal Inventory

• Inventory for time-varying (seasonal) demands
  Trade-off resource acquisition or overtime
  cost -gt optimized in multi-period LND
  Trade-off setup cost -gt optimized in
  Lot-sizing

Demand
Resource Upper Bound
Period
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Cycle Inventory

• Inventory caused by periodic activitiesTrade-off
  transportation fixed cost -gt LNDTrade-off
  ordering fixed cost-gt Economic Ordering Quantity
  (EOQ)

Inventory Level
demand
Cycle Time
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Lot-size Inventory

• Cycle inventory when the speed of demand is not
  constantTrade-off fixed cost -gtLot-sizing,
  multi-period LND

Time
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Safety Inventory

• Inventory for the demand variabilityTrade-off
  customer service level -gtSafety Stock
  Allocation, LNDTrade-off backorder (stock-out)
  cost -gtInventory Policy Optimization

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Classification of Inventory
Seasonal Inventory
Cycle Inventory Lot-size Inventory
Safety Inventory
In-transit (Pipeline) Inventory
Time
Its hard to separate them butThey should be
determined separately to optimize the trade-offs
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Logistics Network Design

• Decision support in strategic level
• Total optimization of overall supply chains

• Example
• Where should we replenish pars?
• In which plant or on which production line should
  we produce products?
• Where and by which transportation-mode should we
  transport products?
• Where should we construct (or close) plants or
  new distribution centers?

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Trade-off in Facility Location Model Number of
Warehouses v.s.

• Service lead time ?
• Inventory cost ?
• Overhead cost ?
• Outbound transportation cost ?
• Inbound transportation cost ?

Number of warehouses
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Trade-off In-transit inventory cost v.s.
Transportation cost
In-transit inventory cost
Transportation cost
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Multi-period logistics network design model

• Decision support in tactical level
• An extension of MPS (Master Production System)
  for production to the Supply Chain
• Treat the seasonal demand explicitly

Demand
Period (Month)
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Trade-offOvertime v.s. Seasonal Inventory Cost
Overtime penalty
Seasonal inventory
Demand
Resource Upper Bound
Period
Overtime
Variable Production
Constant Production
Inventories
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Model MIPConcave Cost Minimization
Safety Stock Cost
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Safety Stock Allocation

• Decision support in tactical level
• Determine the allocation of safety stocks in the
  SC for given service levels

Safety Stock
Service Level
Statistical Economy of Scale or Risk Pooling
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Basic Principle of Inventory

• Economy of scale in statistics gathering
  inventories together reduces the total inventory
  volume.
• -gt Modern supply chain strategies
• risk pooling
• delayed differentiation
• design for logistics

Where should we allocate safety stocks to
minimize the total safety stock costs so that
the customer service level is satisfied.
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Lead-time and Safety Stock

• Normal distribution with average demand
  µ,standard deviation s
• Service level (the probability with no lost
  sales) 95-gtsafety stock ratio 1.65
• Lead-time (the time between ordering and arrival
  of item) L

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The relation between lead-time and
(average,safety,maximum) inventory
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Safety Stock and Guaranteed Lead-time

• Guaranteed Lead-time (LT)Each stocking point
  guarantees to deliver the item to his customer
  within the guaranteed lead-time

Guaranteed LT to downstream point 2 days
Safety stock 2 days
2
2
1
Production time3
Guaranteed LTof upstream point 1 day Entering
LT
Stocking point
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An example Serial multi-stage model
Average demand100 units/day Standard deviation
of demand100 Normal distribution (truncated),
Safety stock ratio1
Guaranteed lead-times of all stocking points 0
Production time 3 days 2 days 1day
1day Inventory cost 10 20 30
40 Safety stock cost 1732
2828 3000 4000
Total 11560
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Optimal Solution
Guaranteed LT3 Entering LT2 Safety
stock3-(21)0 day
Production time 3 days 2 days
1 day 1 day Guaranteed LT 0 day
2 days 3 days 0 day Safety stock
cost 1732 0
0 8000

Total 9732 (16 down)


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Algorithms for Safety Stock Allocation

• Concave cost minimization using piece-wise linear
  approximation
• Dynamic programming (DP) for tree networks
• Metaheuristics(Local Search, Iterated LS, Tabu
  Search)

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A Real Example Ref. Managing the Supply Chain
The Definitive Guide for the Business
Professional by Simchi-Levi, Kaminski,Simchi-Levi

15
x2
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Part 1 Dallas (260)
5
28
Part 2 Dallas (0.5)
Part 4 Malaysia (180)
30
30
15
15
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Final Demand N(100,10) Guaranteed LT 30 days
15
39
3
37
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Part 5 Charleston (12)
Part 3 Montgomery (220)
58
29
37
58
4
8
43,508 (40Down)
Part7 Denver (2.5)
Part 6 Raleigh (3)
What if analysis Guaranteed LT15 days -gt51,136
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Inventory Policy Optimization

• Decision support in operational level
• Determine various parameters for inventory
  control policies

Fixed Ordering
Lost Sales
Safety Stock
Cycle Inventory
Classical Economic Ordering Quantity Model
Classical Newsboy Model
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Economic Ordering Quantity (EOQ) Model

• Given
• d (items/day) a constant demand rate.
• Q (items) order quantities.
• K (yen) a fixed set-up cost of an order.
• h (yen/dayitem) an inventory holding cost per
  item per day.
• Find the optimal ordering policy minimizing total
  ordering and inventory carrying cost over
  infinite planning horizon.

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Inventory
d
Q
Time
Cycle Time (T days)
Cost over T days f(T) Cost per day
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Find the optimal ordering quantify

• Minimize f(T)
• So f(T) is convex. By solving f0, we get

positive
EOQ (Harris) formula
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Newsboy Problem

• inventory cost
• backorder (lost sales) cost
• demand of newspaper (random variable)
• Distribution function of the demand
• Density function

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Expected Value of Total Cost

• Expected cost when the ordering amount is s

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Optimal Solution

• First-order differentiation

Second-order differentiation
is convex!
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Base-stock Policy

• Base stock levelTarget of the inventory position
• Inventory positionIn-hand inventoryIn-transit
  inventory-Backorder
• Base stock policy Monitoring the inventory
  position in real time if it is below the base
  stock level, order the amount so that it recovers
  the base stock level

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Base Stock Policy (Multi Stage Model)

• n serial inventory stocking points
• demand point is 1
• final supplier is n1 that has enough inventory

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Notations (1)

• time index
• local stock at the i-th point
• backorder at the i-th point
• net inventory at the i-th point

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Notations (2)

• inventory on order
• inventory in transit (transit
  inventory)

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Notations (3)
inventory ordering position
inventory transit position
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Notations (4)
lead time

demand between time interval (s,t
base stock level
backorder cost
inventory cost
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Inventory Flow Conservation Equation
base stock level si
By using
ITPi(t)
gtrandom demand
INi(tLi)
Li
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Recursive Equation
equilibrium value of stationary demand during
lead time
Using
i1
i
can compute B from n1 to 1. gtcannot compute
the opt. base stock levels
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Echelon Inventory Model
echelon inventory at the i-th point
system backorder
net echelon inventory
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Echelon Inventory Model
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Notations (Contd)
echelon inventory ordering position
echelon inventory transit position
echelon base stock level at the i-th point

Echelon base stock policy Order the amount so
that the inventory ordering position recovers
the echelon base stock level.
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Echelon Inventory Flow Conservation Equation
echelon inventory cost at the i-th point
Flow conservation equation for echelon inventory
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Recursive Equation
equilibrium value of stationary demand during
lead time
gtcan compute net inventory from n to 1
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Objective Function

• Local inventory model

Echelon inventory model
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Derivation of Optimal Solution (1)
expected cost for 1 to i points when INi1 is x
expected cost for 1 to i points when INi is x
expected cost for 1 to i points when ITPi is y
gtConvex Function
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Derivation of Optimal Solution (2)
expected cost for 1 to i points when INi is x
The minimum cost to the i-1st point when the
echelon net inventory at the i-th point is x
i
i-1
gtLinearConvexConvex
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Derivation of Optimal Solution (3)
expected cost for 1 to i points when ITPi is y
The minimum cost to the i-th point when the
echelon net inventory is y- Di
yITPi
gtrandom demand Di
IN
Li
gt Expectation of convex functions gt convex
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Derivation of Optimal Solution (4)
expected cost for 1 to i points when INi1 is x
i1
i
Echelon net inventory x INi1
y
Minimum cost when
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Derivation of Optimal Solution (5)
C is convex

• Echelon base stock level

Since echelon base stock level is non-decreasing,
The optimum local base stock level
where
is
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Basic Formula of SCM
Is convex
Basic formula of SCM
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(Q,R) and (s,S) Policies

• If the fixed ordering cost is large, the
  ordering frequency must be considered explicitly.
• (Q,R) policyIf the inventory position is below a
  re-ordering point R, order a fixed quantity Q
• (s,S) policyIf the inventory position is below
  a re-ordering point s, order the amount so that
  it becomes an order-up-to level S

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Periodic Ordering Policy

• Check the inventory position periodically if it
  is below the base-stock level, order the amount
  so that it recovers the base-stock level

Order
Mon.
Tue.
Wed.
Thu.
Demand
Arrival of the order of Mon.(Lead-time1day)
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Algorithms for Inv. Policy Opt.

• Base-stock,(Q,R), and (s,S) policies-gtDP
• Periodic ordering policy-gt Infinitesimal
  Perturbation Analysis During simulation runs,
  derivatives of the cost function are estimated
  and are used in non-linear optimization

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Lot-size Optimization

• Decision support in tactical level
• Optimize the trade-off between the set-up cost
  and the lot-size inventory

Setup Cost
Lot-size Inv.
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Basic Single Item Model (1)Parameters

• T Planning horizon (number of periods)
• dt Demand during period t
• ft Fixed order (or production set-up) cost
• ct Per-unit order (or production) cost
• ht Holding cost per unit per period
• Mt Upper bound of production (capacity) in
  period t

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Basic Single Item Model (2)Variables

• It Amount of inventory at the end of period t
  (initial inventory is zero.)
• xt Amount ordered (produced) in period t
• yt 1 if xt gt0, 0 otherwise (0-1 variable),
  i.e. , 1 production is positive, 0 otherwise
  (it is called set-up variable.)

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Basic Single Item Model (3)Formulation
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Lot-sizing (Basic Flow) Model
Production x(t)
Inventory I(t-1)
I(t)
t
Demand d(t)
Week formulation
x(t)? Large M y(t) set-up variable
I(t-1)x(t) d(t)I(t)
0-1 variable
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Valid Inequality
Then the inequality (called the (S,l) inequality)
is valid.
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Valid Inequality,Cut,Facet
Inequality of week formulation (valid inequality)






Facet
Relaxed solution x
Solution x
Integer Polyhedron
Cut
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Extended (Strong) FormulationNotations
Xst ratio of the amount produced in period s to
satisfy demand in period t ( )

The cost produced in period s
to satisfy demand in period t
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Extended (Strong) FormulationFacility Location
Formulation
gt Strong formulation it gives an integer hull
of solutions
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Lot-sizing ModelFacility Location Model
Ratio of the amount produced in period s to
satisfy demand in period t Xst
s
t
d(t)
Xst ?y(t)
Xst 1
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Extended Formulation and Projection
is a formulation of X Q is an extended
formulation of X
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Facility Location Formulation and Projected
Polyhedron
Extended Formulation (Facility Location
Formulation)
Projection
Integer Polyhedron of Original Formulation
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Comparison of Size and Strength
Standard Formulation
Facility Location Formulation
of var.s
of var.s
of const.s
Week formulation
Strong formulation linear prog. relax. integer
polyhedron
of const.s
(S, l) ineq.s cut
added const.s
T of periods
Strong formulation
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Dynamic Programming for the Uncapacitated Problem
Upper bound of production (capacity) Mt is
large enough.
F(j) Minimum cost over the first j periods
(F(0)0)
O(T2) or O(T log T) time algorithm
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Silver-Meal Heuristics
Define
Let t1. Determine the first period j (gtt) that
satisfies (If such j does not exist, let jT.)
The lot size produced in period t is the total
demand from t to j. Let tj1 and repeat the
process until jT.
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Least Unit Cost Heuristics
Let t1. Determine the first period j (gtt) that
satisfies (If such j does not exist, let
jT.) The lot size produced in period t is the
total demand from t to j. Let tj1 and repeat
the process until jT.
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Example Single Item Model
Period (day,week,month,hour)1,2,3,4,5 (5 days)
production
setup
Setup cost 3 demand 5,7,3,6,4
(tons) Inventory cost 1 per day Production
cost 1,1,3,3,3 per ton
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Comparison with ad hoc methods
Product at once setup (3)production(25)invento
ry(2013104)75
Just-in-time productionsetup(15)prod.(51)inv.(0
)66
Optimal productionsetup(9)prod.(33)inv.(15)57
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Comparison with heuristics
Silver-Meal heuristics Determine the lot-size so
that the cost per period is minimized.
setup(9)prod.(45)inventory(7)61
Least unit cost heuristics Determine the lot-size
so that the cost per unit-demand is minimized.
setup(9)prod(51)inventory(14)74
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Algorithms for Lot-sizing

• Metaheuristics using MIP solver
• Relax and Fix
• Capacity scaling
• MIP neighbor local search

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Scheduling Optimization

• Decision support in operational level
• Optimization of the allocation of activities
  (jobs, tasks) over time under finite resources

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What is the scheduling?

• Allocation of activities (jobs, tasks) over time
• Resource constraints. For example, machines,
  workers, raw material, etc. may be scare
  resources.
• Precedence relation. For example., some
  activities cannot start unless other activities
  finish.

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Solution methods for scheduling

• Myopic heuristics
• Active schedule generation scheme
• Non-delay schedule generation scheme
• Dispatching rules
• Constraint programming
• Metaheuristics

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Vehicle Routing Optimization
earliest time
latest time
Customer
service time
waiting time
service time
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Algorithms for Vehicle Routing

• Saving (Clarke-Wright) method
• Insertion method
• Guided Local Search
• Iterated Local Search

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History of Algorithms for Vehicle Routing Problem
Approximate Algorithm
Genetic Algorithm
AMP (Adaptive Memory Programming)
Tabu Search
Local Search
Simulated Annealing
Sweep Method
Generalized Assignment
Location Based Heuristics
Route Selection Heuristics
GRASP (Greedy Randomized Adaptive Search
Procedure)
Construction Method (Saving, Insertion)
Hierarchical Building Block Method
Exact Algorithm
Set Partitioning Approach
State Space Relax.
Cutting Plane
K-Tree Relax.
1970
1980
1990
2000
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Conclusion

• Decision Levels of SC
• Classification of Inventory
• Basic Models in SC
• Logistics Network Design
• Inventory
• Production Planning
• Vehicle Routing