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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