Agenda. Real System, Transactional IT, Analytic IT. What s the Supply Chain. Levels of Decision Making. Supply Chain Optimization

Transcription

1 Agenda Supply Chain Optimization KUBO Mikio Definition of the Supply Chain (SC) and Logistics Decision Levels of the SC Classification of Basic Models in the SC Logistics Network Design Production Planning Vehicle Routing 1 2 What s the Supply Chain Real System, Transactional IT, Analytic IT brain 解 析 的 IT Analytic IT Model+Algorithm= Decision Support System IT(Information Technology)+Logistics =Supply Chain 3 実 システム nerve 処 理 的 IT Transactional IT POS, ERP, MRP, DRP Automatic Information Flow Real System=Truck, Ship, Plant, Product, Machine, muscle 4 Levels of Decision Making Models in Analytic IT Analytic IT Strategic Level A year to several years; long-term decision making Tactical Level A week to several months; mid-term decision making Supplier Strategic Plant DC Retailer Logistics Network Design Multi-period Logistics Network Design Transactional IT Operational Level Real time to several days; short-term decision making 5 Tactical Operational Safety stock allocation policy optimization Production Planning Lot-sizing Scheduling Transportation Delivery Vehicle Routing 6 1

2 Models in Analytic IT Models in Analytic IT Supplier Plant DC Retailer Supplier Plant DC Retailer Strategic Logistics Network Design Strategic Logistics Network Design Tactical Operational Safety stock allocation policy optimization Multi-period Logistics Network Design Production Planning Lot-sizing Scheduling Transportation Delivery Vehicle Routing Tactical Operational Multi-period Logistics Network Design Production Transportation Planning Delivery Safety stock allocation policy optimization Lot-sizing Scheduling Vehicle Routing 7 8 =Blood of Supply Chain acts as glue connecting optimization systems Classification of In-transit (pipeline) inventory Trade-off: transportation cost or production speed Supplier Plant DC Retailer Seasonal inventory Trade-off: resource acquisition or overtime cost,setup cost Raw material Work-in-process Finished goods Cycle inventory Trade-off : transportation (or production or ordering) fixed cost Lot-size inventory Trade-off: fixed cost 9 Safety inventory Trade-off: customer service level, backorder (stockout) cost 10 In-transit (pipeline) Seasonal that are in-transit of products Trade-off: transportation cost or transportation/production speed ->optimized in Logistics Network Design (LND) 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 11 Period 12 2

3 Cycle Lot-size caused by periodic activities Trade-off : transportation fixed cost -> LND Cycle inventory when the speed of demand is not constant Trade-off: ordering fixed cost -> Economic Ordering Quantity (EOQ) Trade-off: fixed cost ->Lot-sizing, multi-period LND Level demand Level Cycle Safety for the demand variability Classification of Cycle Lot-size Seasonal Trade-off: customer service level ->Safety Stock Allocation, LND Trade-off: backorder (stock-out) cost -> Policy Optimization Safety In-transit (Pipeline) 15 It s hard to separate them but 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. Number 輸 送 of 中 在 warehouses 庫 費 用 Service lead time cost Overhead cost Outbound 輸 送 費 用 transportation cost Inbound transportation cost 18 3

4 Trade-off: In-transit inventory cost v.s. Transportation cost In-transit 輸 送 中 inventory 在 庫 費 用 cost Transportation 輸 送 費 用 cost 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 19 Period (Month) 20 Trade-off: Overtime v.s. Seasonal Cost Demand Resource Upper Bound Overtime 資 源 超 過 penalty ペナルティSeasonal 作 り 置 き inventory 在 庫 費 用 ( 残 業 費 ) Mixed Integer Programming (MIP) + Concave Cost Minimization BOM or orrecipe Recipie 3 Safety Inv. Cost Warehouses Customer Gropus Period Overtime Seasonal Constant Production Variable Production Suppliers Plants Production Lines MIP Formulation of Simple Facility Location Problem transportation costs from plants to customers fixed costs of plants Safety Stock Allocation Decision support in the tactical level Determine the allocation of safety stocks in the SC for given service levels Safety 安 全 在 庫 費 用 サービスレベル Service Level =1 if the plant is open, =0 otherwise transportation volume from plants to customer 23 + 統 計 的 +Risk 規 模 の Pooling 経 済 (リスク (Statistical 共 同 Economy 管 理 ) of Scale) 24 4

5 Basic Principle of 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= μ L +Safety Stock Ratio σ L 26 The Relation between Lead-time and (Average, Safety, Maximum) Average Max. Safety Guaranteed Lead-time Guaranteed lead-time (LT):Each facility guarantees to deliver the item to his customer within the guaranteed lead- time Safety inv. =2 days Guaranteed LT to downstream facility L i =2 days Lead-time 27 Guaranteed LT of upstream facility =1 day = Entering LT LI i Production time T i =3 Facility i 28 Net Replenishment Net replenishment time (NRT): =LT i +T i -L i Guaranteed LT of upstream facility =1 day = Entering LT LI i Safety inv. =2 days 2 1 Production time T i =3 Facility i Guaranteed LT to downstream facility L i =2 days 2 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 cost per unit 10$ 20$ 30 $ 40 $ Safety inv. cost 1732 $ 2828 $ 3000 $ 4000 $ Total $ 30 5

6 Optimal Solution Further Improvement Guaranteed LT=3 Entering LT=2 Safety stock=2+1-3=0 day Safety stock cost is decreased from 9732$ to 6928$ by increasing the guaranteed lead time to the customer from 0 to 1. 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 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 Safety Stock Allocation Formulation maximum demand maximum demand net replenishment time net replenishment time minimum cost from facility n to stage i when the guaranteed LT of facility i is L i : initial condition 33 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 Part 4 Malaysia ($180) Part7 Denver ($2.5) 37 3 Part 5 Charleston ($12) Part x2 Part 1 Dallas ($260) Part 2 Dallas ($0.5) Final Demand Montgomery ($220) N(100,10) Guaranteed LT =30 days 43,508$ (40%Down) Part 6 Raleigh ($3) What if analysis: Guaranteed LT=15 days ->51,136$ 36 6

7 Policy Optimization Decision support in the operational/tactical level Determine various parameters for inventory control policies Lost 安 全 在 Sales 庫 費 用 Fixed Ordering 発 Cycle 注 ( 生 産 ) 固 定 費 用 Safety 品 切 れ 費 用 サイクル 在 庫 費 用 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. Classical Newsboy Model Classical Economic Ordering Quantity Model level d Optimal Ordering Quantify Minimize f(t) Q positive So f(t) is convex. By solving f =0, we get: Cycle (T days) Cost over T days = f(t) = Cost per day = 39 EOQ (Harris ) formula 40 Newsboy Model Expected Value of Total Cost inventory cost lost sales cost demand of newspaper (random variable) Expected cost when the ordering quantity is s : Distribution function of the demand Density function

8 Optimal Solution First-order derivative: Base Stock Policy (Multi Period Model) Second-order order derivative : is convex Base stock level s* = target of the inventory position (ordering) position= In-hand inventory+in-transit i 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 critical ratio Base Stock Policy Lead time Base stock level = position (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 R+Q (=S) (Q,R) Policy and (s,s) Policy position (Q,R) (s,s) Lot-size Optimization Decision support in the tactical level Optimize the trade-off between set-up cost and lot-size inventory R (=s) Lot-size 段 取 り 費 Inv. 用 Setup 在 庫 費 Cost 用 In-hand inventory Lead time

9 Basic Single Item Model (1) Parameters T : Planning horizon (number of periods) d t : Demand during period t f t : Fixed order (or production set-up) cost c t : Per-unit order (or production) cost h t : Holding cost per unit per period M : t Upper bound of production (capacity) in period t Basic Single Item Model (2) Variables I t : Amount of inventory at the end of period t (initial inventory is zero.) x t : Amount ordered (produced) in period t y t : =1 if x t >0, =0 otherwise (0-1 variable), i.e., =1 production is positive, =0 otherwise (it is called setup variable. ) Basic Single Item Model (3) Formulation Lot-sizing (Basic Flow) Model I t-1 Production x t t I t Demand d t Weak formulation x t Large M y t [set-up variable] 51 I t-1 + x t = d t + I t 0-1 variable 52 Valid Inequality Valid Inequality,Cut,Facet Inequality of week formulation (valid inequality) Facet Relaxed solution x* Then the inequality (called the (S,l) inequality) Solution x Integer Polyhedron (Integer Hull) Cut is valid

10 Extended (Strong) Formulation for Uncapacitated Case Upper bound of production (capacity) M t is large enough. X st : ratio of the amount produced in period s to satisfy demand in period t ( ) Lot-sizing Model Facility Location Formulation Ratio of the amount produced in period s to satisfy demand in period t X st s t The cost produced in period s to satisfy demand in period t 55 X st y t s t X st = 1 d t 56 Extended Formulation Facility Location Formulation Extended Formulation and Projection => Strong formulation; it gives an integer polyhedron of solutions is a formulation of X = Q is an extended formulation of X 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 ) 2 # of var.s O( T ) # of const.s s Week O(T ) formulation # of const.s Strong formulation added const.s O( 2 T ) (S, l) ineq.s cut Strong formulation O( T 2 ) linear prog. relax. =integer polyhedron T: # of periods 60 10

11 Dynamic Programming for the Uncapacitated Problem Upper bound of production (capacity) M t is large enough. Silver-Meal Heuristics Define: F(j) : Minimum cost over the first j periods (F(0)=0) Let t=1. Determine the first period j (>=t) that satisfies: O(T 2 ) or O(T log T) time algorithm 61 (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: Example: Single Item Model Period (day,week,month,hour):1,2,3,4,5 (5 days) (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 setup production Setup cost: 3$ demand : 5,7,3,6,4 (tons) 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( )=75 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 Just-in-time production:setup(15)+prod.(51)+inv.(0)=66 Optimal production:setup(9)+prod.(33)+inv.(15)=57 Least unit cost heuristics Determine the lot-size so that the cost per unit-demand is minimized. setup(9)+prod(51)+inventory(14)=

12 Algorithms for Lot-sizing Metaheuristics using MIP solver Relax and Fix Capacity scaling MIP based neighborhood search Scheduling Optimization Decision support in the operational level Optimization of the allocation of activities (jobs, tasks) over time under finite resources (such as machines) Machine 1 Machine 2 Machine 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. Machine 1 Machine 2 Machine Solution Methods for Scheduling Myopic heuristics Active schedule generation scheme Non-delay schedule generation scheme Dispatching rules Constraint programming Metaheuristics Vehicle Routing Optimization Algorithms for Vehicle Routing Customers Depot Routes earliest time waiting time Customer service time latest time Saving (Clarke-Wright) method Sweep (Gillet-Miller) method Insertion method Local Search Metaheuristics service time

13 History of Algorithms for Vehicle Routing Problem Approximate Algorithm Genetic Algorithm Local Search Tabu Search AMP Simulated Annealing (Adaptive Memory Programming) Sweep Generalized Location Based Route Selection Method Assignment Heuristics Heuristics Construction Method (Saving, Insertion) GRASP (Greedy Randomized Adaptive Search Procedure) Exact Algorithm Set Partitioning Approach State Space Relax. Cutting Plane K-Tree Relax Hierarchical Building Block Method 73 Conclusion Definition of the Supply Chain (SC) and Logistics Decision Levels of the SC Classification of Basic Models in the SC Logistics Network Design Production Planning Vehicle Routing 74 13