This formulation can be shown to be tight by D3YeC36

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									Supply Chain Optimization




                  KUBO Mikio



                               1
Agenda
  Definition of the Supply Chain (SC) and
 Logistics
  Decision Levels of the SC
  Classification of Inventory
  Basic Models in the SC
     Logistics Network Design
     Inventory
     Production Planning
     Vehicle Routing


                                            2
What’s the Supply Chain




IT(Information Technology)+Logistics
                         =Supply Chain
                                         3
Real System, Transactional IT,
Analytic IT

                            Analytic IT
                brain
                解析的IT       Model+Algorithm=
                            Decision Support System



                      Transactional IT
                      POS, ERP, MRP, DRP…
           nerve
           処理的IT      Automatic Information Flow


      Real System=Truck, Ship, Plant, Product, Machine, …
 muscle
 実システム                                                      4
   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



Transactional IT
                  Operational Level
              Real time to several days;
                         short-term decision making
                                                                 5
     Models in Analytic IT

Supplier                Plant                      DC               Retailer



 Strategic                      Logistics Network Design

                            Multi-period Logistics Network Design

   Tactical        Inventory             Production
                                                            Transportation
                                          Planning
                                                               Delivery
              Safety stock allocation
                Inventory policy          Lot-sizing        Vehicle Routing
Operational        optimization           Scheduling

                                                                       6
      Models in Analytic IT

Supplier                   Plant                      DC               Retailer



 Strategic                  Logistics Network Design
                               Multi-period Logistics Network Design

   Tactical           Inventory             Production
                                                               Transportation
                                             Planning
                                                                  Delivery
                 Safety stock allocation
                   Inventory policy          Lot-sizing        Vehicle Routing
                      optimization           Scheduling
   Operational
                                                                          7
      Models in Analytic IT

Supplier                 Plant                       DC              Retailer



   Strategic                        Logistics Network Design

                             Multi-period Logistics Network Design
   Tactical                               Production
                 Inventory                                 Transportation
                                           Planning           Delivery
               Safety stock allocation
 Operational     Inventory policy           Lot-sizing
                                            Scheduling         Vehicle Routing
                    optimization
                                                                          8
  Inventory=Blood of Supply Chain
   Inventory acts as glue connecting optimization systems


Supplier             Plant               DC            Retailer


   Raw material       Work-in-process         Finished goods




                                                     Time
                                                               9
Classification of Inventory
  In-transit (pipeline) inventory
  Trade-off: transportation cost or production speed

  Seasonal inventory
  Trade-off: resource acquisition or overtime cost,setup
  cost

  Cycle inventory
  Trade-off : transportation (or production or ordering)
  fixed cost

  Lot-size inventory
  Trade-off: fixed cost

  Safety inventory
  Trade-off: customer service level, backorder (stock-
  out) cost                                                10
In-transit (pipeline) Inventory
 Inventory that are in-transit of products
 Trade-off: transportation cost or
 transportation/production speed
 ->optimized in Logistics Network Design (LND)




                                                 11
Seasonal Inventory
 Inventory for time-varying (seasonal) demands
 Trade-off: resource acquisition or overtime cost
  -> optimized in multi-period LND
 Trade-off: setup cost
 -> optimized in Lot-sizing
                              Demand   Resource Upper Bound




                                                  Period
                                                       12
Cycle Inventory
 Inventory caused by periodic activities

 Trade-off : transportation fixed cost -> LND

 Trade-off: ordering fixed cost
 -> Economic Ordering Quantity (EOQ)
                 Inventory       demand
                 Level




                             Cycle Time         13
 Lot-size Inventory
    Cycle inventory when the speed of
    demand is not constant

    Trade-off: fixed cost
     ->Lot-sizing, multi-period LND
Inventory
Level




                                      Time   14
Safety Inventory
 Inventory for the demand variability

 Trade-off: customer service level
 ->Safety Stock Allocation, LND

 Trade-off: backorder (stock-out) cost
 ->Inventory Policy Optimization



                                         15
    Classification of Inventory
Cycle Inventory                                   Seasonal Inventory
Lot-size Inventory




                      Safety Inventory


                In-transit (Pipeline) Inventory

 It’s hard to separate them but…                         Time
 They should be determined separately to optimize the trade-offs
                                                                  16
Logistics Network Design
   Decision support in the strategic level
   Total optimization of overall supply chains

Example
 Where should we replenish parts?
 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?
                                                 17
Trade-off in LND Model:
Number of Warehouses v.s.
                          •   Service lead time ↓
                          •   Inventory cost ↑
                          •   Overhead cost ↑
   Number of warehouses
      輸送中在庫費用             •   Outbound transportation cost ↓
                               輸送費用
                          •   Inbound transportation cost ↑




                                                          18
Trade-off:
In-transit inventory cost v.s. Transportation cost
   In-transit inventory cost
         輸送中在庫費用                  輸送費用
                               Transportation cost




                                                     19
Multi-period Logistics Network Design
Decision support in the tactical level
An extension of MPS (Master Production System) for
production to the Supply Chain
Treat the seasonal demand explicitly


Demand




                                       Period (Month)
                                                    20
    Trade-off:
    Overtime v.s. Seasonal Inventory Cost
                                   Overtime penalty Seasonal inventory
                                      資源超過ペナルティ 作り置き在庫費用
                                      (残業費)
      Demand
                   Resource Upper Bound




                        Period                      Overtime

                                                           Variable
                     Constant                              Production
                     Production
Seasonal
Inventory




                                                                        21
Mixed Integer Programming (MIP) +
Concave Cost Minimization
     BOM or Recipe
      BOM or Recipie
        ×   3
                          Safety Inv. Cost

                          Warehouses Customer Gropus




                Plant s
Suppliers
                           Product ion Lines
                                                       22
  MIP Formulation of Simple
  Facility Location Problem
                 transportation costs from plants to customers
  fixed costs of plants




                 =1 if the plant is open, =0 otherwise
transportation volume from plants to customer
                                                                 23
Safety Stock Allocation

   Decision support in the tactical level
   Determine the allocation of safety
   stocks in the SC for given service levels
Safety Inventory
  安全在庫費用            Service Level
                   サービスレベル




                         +統計的規模の経済+Risk Pooling
                         (Statistical Economy
                         (リスク共同管理) of Scale)



                                               24
Basic Principle of Inventory
  Economy of scale in statistics: gathering
  inventory together reduces the total
  inventory volume.
 -> 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.
                                                           25
Lead-time and Safety Stock
  Normal distribution with average demand μ,
  standard deviation σ
  Service level (the probability of no stocking
  out) 95%->safety stock ratio 1.65
  Lead-time (the time between order and
  arrival) L

Max Inv.Volume=
  L+Safety Stock Ratio    L
                                                  26
The Relation between Lead-time and
(Average, Safety, Maximum) Inventory
 3000

 2500

 2000
                                      Average
 1500                                 Max.
                                      Safety
 1000

 500

    0
        0   5      10       15   20
                Lead-time

                                                27
    Guaranteed Lead-time
         Guaranteed lead-time (LT):Each facility
         guarantees to deliver the item to his
         customer within the guaranteed lead-
         time                 Guaranteed LT to
                        Safety inv.        downstream facility
                        =2 days            Li =2 days
                               2            2
Guaranteed LT
of upstream facility      1   Production time   Ti =3
=1 day
= Entering LT     LIi         Facility i
                                                                 28
    Net Replenishment Time
         Net replenishment time (NRT):
        =LTi +Ti -Li

                                           Guaranteed LT to
                        Safety inv.        downstream facility
                        =2 days            Li =2 days
                               2            2
Guaranteed LT
of upstream facility      1    Production time   Ti=3
=1 day
= Entering LT     LIi         Facility i
                                                                 29
    Example: Serial Multi Stage Model
                   Average demand=100 units/day
                   Standard deviation of demand=100
                   Normal distribution (truncated), Safety stock ratio=1
         Guaranteed lead-times of all stocking points =0

                                                           Customer

    Pars Maker      Plant     Wholesaler     Retailer
Production time
          3 days    2 days      1day          1day
Inventory cost per unit
         10$         20$         30 $         40 $
Safety inv. cost
       1732 $       2828 $     3000 $       4000 $ Total 11560 $
                                                                      30
   Optimal Solution
                                Guaranteed LT=3
                                Entering LT=2
                                Safety stock=2+1-3=0 day




Production time                            push       pull
              3 days   2 days     1 day       1 day
Guaranteed LT
              0 day    2 days     3 days      0 day
Safety inv. cost
          1732 $        0$          0$       8000$
                                     Total 9732$ (16% down)   31
   Further Improvement
     Safety stock cost is decreased from 9732$ to 6928$ by increasing the
     guaranteed lead time to the customer from 0 to 1.




                   push                                pull
Production time
              3 days       2 days      1 day          1 day
Guaranteed LT
              0 day        2 days      0 day          1 day
Safety inv. cost
          1732 $             0$         5196 $       0$
                                           Total 6928$ (40 % down)          32
Serial Multi Stage Safety Stock Allocation
Dynamic Programming

                                            maximum demand




                                           net replenishment time
 minimum cost from facility n to stage i
        when the guaranteed LT of facility i is Li

                              : initial condition
                                                                33
Safety Stock Allocation
Formulation       maximum demand



                        net replenishment time




                upper bound of guaranteed LT
                                                 34
Algorithms for Safety Stock Allocation

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


                                            35
   A Real Example: Ref.               Managing the Supply Chain –The
   Definitive Guide for the Business Professional –by Simchi-Levi,
   Kaminski,Simchi-Levi

                                                       15 x2
                             37                           5       Part 1
                               28                                 Dallas ($260)
                                               Part 2
                     Part 4
                                               Dallas ($0.5)        30
                     Malaysia ($180)                                          30
                                                                  15 15
                             37                 39     15         Final Demand
                                  3                               N(100,10)
                      Part 5
                                               37       17
                                                                  Guaranteed LT
                      Charleston ($12)        Part 3              =30 days
                                              Montgomery ($220)
   58                     29          37
        4                 58          8
                                                   43,508$ (40%Down)
Part7                    Part 6
Denver ($2.5)            Raleigh ($3)
                                           What if analysis:
                                           Guaranteed LT=15 days ->51,136$         36
     Inventory Policy Optimization

         Decision support in the operational/tactical
         level
         Determine various parameters for inventory
         control policies
                   Lost Sales
                   安全在庫費用                           Fixed Ordering
                                                      サイクル在庫費用
Safety Inventory
 品切れ費用                           Cycle Inventory
                                発注(生産)固定費用




  Classical Newsboy Model            Classical Economic Ordering
                                     Quantity Model              37
Economic Ordering Quantity (EOQ)
 Given
     d : constant demand rate
     Q : order quantity
     K : fixed set-up cost of an order
     h : inventory holding cost per item per day
 Find the optimal ordering policy minimizing
 total ordering and cycle inventory cost over
 infinite planning horizon.



                                                    38
Inventory
level

                 d


      Q




        Cycle Time (T days)       Time
        Cost over T days =

          f(T) = Cost per day =
                                         39
Optimal Ordering Quantify

  Minimize f(T)

                                               positive


  So   f(T) is convex. By solving f’=0, we get:




                           EOQ (Harris’) formula
                                                     40
Newsboy Model
  inventory cost
  lost sales cost
  demand of newspaper (random variable)

Distribution function of the demand



Density function

                                          41
Expected Value of Total Cost
Expected cost when the ordering quantity is s :




                                                  42
 Optimal Solution
   First-order derivative:




Second-order derivative :


       is convex


                             critical ratio
                                              43
Base Stock Policy (Multi Period Model)

  Base stock level s* = target of the
  inventory position
  Inventory (ordering) position=
  In-hand inventory+In-transit inventory
  (inventory on order) -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

                                                   44
Base Stock Policy   Base stock level
                    =Inventory position




    Lead time


                                  Time
                                          45
(Q,R) and (s,S) Policies
   If the fixed ordering cost is positive, the
  ordering frequency must be considered
  explicitly.
  (Q,R) policy:If the inventory position is below
  a re-ordering point R, order a fixed quantity Q
   (s,S) policy:If the inventory position is below
  a re-ordering point s, order the amount so that
  it becomes an order-up-to level S


                                                     46
   (Q,R) Policy and (s,S) Policy
R+Q
(=S)              Inventory           (s,S)
                  position
                              (Q,R)




R
(=s)

                         In-hand
                         inventory

            Lead time                         Time
                                                     47
Lot-size Optimization
Decision support in the tactical level
Optimize the trade-off between set-up cost and lot-size
inventory


                      Lot-size Inv.
                      段取り費用                   Setup Cost
                                              在庫費用




                                                           48
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


                                                 49
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 >0, =0 otherwise (0-1
  variable), i.e. , =1 production is
  positive, =0 otherwise (it is called “set-
  up variable.”)

                                               50
Basic Single Item Model (3)
Formulation




                              51
Lot-sizing (Basic Flow) Model
           Production
              xt
  Inventory
    It-1                It
               t


       Demand      dt           Weak formulation

        xt ≦ “Large M” × yt [set-up variable]
  It-1 + xt = dt + It                     0-1 variable
                                                         52
 Valid Inequality




Then the inequality (called the (S,l) inequality)




is valid.
                                                    53
     Valid Inequality,Cut,Facet
Inequality of week formulation
(valid inequality)            Facet

                                      Relaxed solution x*


Solution x
                                      Integer Polyhedron
                                      (Integer Hull)

                                                Cut




                                                            54
Extended (Strong) Formulation
for Uncapacitated Case
 Upper bound of production (capacity) Mt is large enough.

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

                                                            55
Lot-sizing Model
Facility Location Formulation
                Ratio of the amount produced
                in period s to satisfy demand in period t
                Xst

         s                            t


                                     dt
        Xst ≦ yt
          Xst = 1
         s t                                           56
 Extended Formulation
 Facility Location Formulation




=> Strong formulation; it gives an integer polyhedron of solutions




                                                                     57
Extended Formulation and
Projection




   is a formulation of X
   = Q is an extended formulation of X


                                         58
Facility Location Formulation and
Projected Polyhedron
                        Extended Formulation
                        (Facility Location Formulation)




                                    Projection




                    Integer Polyhedron
                     of Original Formulation         59
    Comparison of Size and
    Strength
           Standard Formulation              Facility Location Formulation
           # of var.s   O (T )                   # of var.s O(T 2 )
# of
 const.s           Week
 O (T )         formulation       # of const.s   Strong formulation
                                         2
added
                                   O(T )          linear prog. relax.
                (S, l) ineq.s                    =integer polyhedron
 const.s
     T
O(2 )               cut
                                                    T: # of periods
            Strong formulation
                                                                             60
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

                                                           61
  Silver-Meal Heuristics
 Define:




Let t=1. Determine the first period j (>=t) that satisfies:




(If such j does not exist, let j=T.) The lot-size produced in
period t is the total demand from t to j. Let t=j+1 and repeat the
process until j=T.
                                                                     62
Least Unit Cost Heuristics
Let t=1. Determine the first period j (>=t) that
satisfies:




(If such j does not exist, let j=T.) The lot size
produced in period t is the total demand from t to j.
Let t=j+1 and repeat the process until j=T.


                                                        63
Example: Single Item Model
Period (day,week,month,hour):1,2,3,4,5 (5 days)




    setup             production

Setup cost: 3 $
demand      : 5,7,3,6,4 (tons)
Inventory cost : 1 $ per day
Production cost : 1,1,3,3,3 $ per ton
                                                  64
Comparison (1): Ad Hoc Methods
 Product at once:
  setup (3)+production(25)+inventory(20+13+10+4)=75




 Just-in-time production:setup(15)+prod.(51)+inv.(0)=66




 Optimal production:setup(9)+prod.(33)+inv.(15)=57



                                                          65
Comparison (2) : 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




                                                              66
Algorithms for Lot-sizing
  Metaheuristics using MIP solver
     Relax and Fix
     Capacity scaling
     MIP based neighborhood search




                                      67
Scheduling Optimization

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

                                                 Time
            0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Machine 1
Machine 2
Machine 3

                                                        68
What is 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.
                                                    Time
            0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Machine 1
Machine 2
Machine 3

                                                            69
Solution Methods for Scheduling
  Myopic heuristics
     Active schedule generation scheme
     Non-delay schedule generation scheme
     Dispatching rules


  Constraint programming

  Metaheuristics


                                             70
  Vehicle Routing Optimization

Customers
                             earliest time             latest time
                                             Customer



            Depot

                              waiting   service time
                               time

                    Routes
                                                service time



                                                                 71
Algorithms for Vehicle Routing
  Saving (Clarke-Wright) method
  Sweep (Gillet-Miller) method
  Insertion method
  Local Search
  Metaheuristics




                                  72
  History of Algorithms for Vehicle Routing
  Problem
Approximate Algorithm        Genetic Algorithm
                  Tabu Search       AMP
Local Search     Simulated Annealing       (Adaptive Memory
                                            Programming)
Sweep     Generalized        Location Based      Route Selection
Method    Assignment         Heuristics          Heuristics

Construction Method     GRASP
                        (Greedy Randomized
(Saving, Insertion)     Adaptive Search Procedure)        Hierarchical
                                                          Building Block
 Exact Algorithm    Set Partitioning Approach             Method
             State Space Relax.      Cutting Plane
                        K-Tree Relax.
    1970        1980      1990              2000
                                                                      73
Conclusion
  Definition of the Supply Chain (SC) and
 Logistics
  Decision Levels of the SC
  Classification of Inventory
  Basic Models in the SC
     Logistics Network Design
     Inventory
     Production Planning
     Vehicle Routing



                                            74

								
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