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Production Inventory Networks

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    Production Inventory Networks Production Inventory Networks Presentation Transcript

    • Production Inventory Networks: Perspectives and Recent Advances
      • Saif Benjaafar
      • Industrial & Systems Engineering Division
      • Department of Mechanical Engineering
      • University of Minnesota
      Presented at Chinese Academy of Sciences, June 16, 2007
      • Benjaafar, S. and M. ElHafsi, “Production and Inventory Control of an ATO System with Multiple Customer Classes,” Management Science , 51 , 2006
      • Benjaafar, S., Y. Li and D. Xu, “Demand Allocation in Systems with Multiple Inventory Locations and Multiple Demand Sources,” M&SOM , forthcoming, 2007
      • Gayon, J. P., S. Benjaafar and F. de Véricourt, “Using Imperfect Demand Information in Production-Inventory Systems with Multiple Demand Classes,” M&SOM , forthcoming, 2007
      • Benjaafar, S., William L. Cooper and J. S. Kim, “On the Benefits of Pooling in Production-Inventory Systems,” Management Science , 51 , 548-565, 2005
      • Benjaafar, S., M. Elhafsi and F. de Véricourt, “Demand Allocation in Multi-Product, Multi-Facility MTS Systems,” Management Science , 50 , 1431–1448, 2004
      Some Recent Papers
      • Benjaafar, S., E. Elahi and K. Donohue, “Outsourcing via Service Quality Competition,” Management Science , 53 , 241-259, 2007.
      • Benjaafar, S., J. S. Kim and N. Vishwanadham, “On the Effect of Product-Variety in Production-Inventory Systems,” Annals of Operations Research , 126 , 71-101, 2004
      • Gupta, D. and S. Benjaafar, “Make-to-order, Make-to-stock, or Delay Product Differentiation? - A Common Framework for Modeling and Analysis,” IIE Transactions , 36 , 529-546, 2004
      • Benjaafar, S., M. ElHafsi, C. Y. Lee and W. Zhou, “Optimal Control of Assembly Systems with Multiple Stages and Multiple Demand Classes,” Working Paper, 2006
      • Benjaafar, S., W. L. Cooper and S. Mardan, “Production-Inventory Systems with Imperfect Advance Demand Information and Due-Date Updates,” Working Paper, 2006
      Some Recent Papers (Continued…)
      • Examples of production-inventory systems
      • Issues in design, analysis and control
      • Multi-stage assembly systems
      • Systems with imperfect advance demand information
      • Ongoing and future work
        • Production & Inventory Systems
        • Service Systems
      Outline
    • A Production-Inventory System Production facility Finished-goods inventory Customer orders Raw Materials Customer shipments Production orders
    • Characteristics
      • Tight coupling of production & inventory
      • Limited production capacity, items produced one unit at a time
      • Variability in both demand and production times
      • Supply lead times are affected by congestion at the production system
    • Inventory Literature
      • Production and inventory treated separately
      • Capacity constraints are frequently ignored
      • Supply leadtimes are not affected by congestion
    • Issues
      • How much inventory should we keep (should we produce to order or to stock)?
      • When should we place a production order and for how much?
      • When should we initiate production and for how long?
      • What is the impact of system parameters?
    • A Series System Raw Materials Production orders Customer orders Production orders Production orders Customer shipments Supply orders Supply orders
    • Issues
      • Where should we keep inventory and how
      • much?
      • How should we coordinate production across
      • stages?
      • Where should we invest in capacity and in variability reduction?
    • A Distribution System Raw Materials Customer orders from location 1 Customer orders from location 2 Customer orders from location N
    • Issues
      • How much inventory to stock of each product (which product to make to order and which to make to stock)?
      • How should priorities be assigned to different products?
      • What is the impact of inventory consolidation?
    • An Assemble-to-Order System Product 1 Product N Product 2 Production facilities Components
    • Issues
      • How much of each component to stock?
      • How should we coordinate the production of different components?
      • How should shared components be allocated?
      • What is the impact of various parameters?
    • A Multi-Stage Assembly System
      • Benjaafar et al. (MS, 2005; AOR, 2004): Impact of inventory pooling in a distribution system
      • Benjaafar et al. (MS, 2006): Optimal control of ATO systems
      • Benjaafar et al. (MS, 2004; MSOM 2007): Demand allocation in a network with multiple production facilities/inventory locations
      • Benjaafar et al. (M&SOM, 2007): Optimal control of systems with ADI
      • Benjaafar et al. (IIE Transactions, 2004): MTS versus MTO systems
      • Benjaafar et al. (MS, 2007): Competition among MTS suppliers
      Related Papers
    • Benjaafar et al. (AOR 2004, MS 2005) Raw Materials Customer orders from location 1 Customer orders from location 2 Customer orders from location N
    • Benjaafar and Gupta (IIE Transactions, 2004) Raw Materials Customer orders Customer shipments Make-to-stock segment Make-to-order segment
    • Benjaafar et al. (MS, 2004; MSOM 2007)
    • Benjaafar and ElHafsi (MS 2006) Production facilities Components
    • Benjaafar et al. (MS, 2007) Competing suppliers A buyer that allocates demand
    • Benjaafar et al. (MSOM, 2007) Orders are announced Demand leadtime process Orders are due
      • Benjaafar et al. (2007): Multi-stage assembly systems with multiple stages with multiple classes
      • Gayon et al. (2006): Systems with imperfect advance demand information
      • Benjaafar et al. (2006): Systems with imperfect advance demand information and due-date update
      Papers Related to this Talk
    • Optimal Control of Multi-Stage Assembly Systems with Multiple Demand Classes (Joint work with Mohsen Elhafsi, Larry Zhou and Chung-Yee Lee )
    • A Multi-Stage Assembly System
    • Motivation
      • Assembly permeates most manufacturing and multi-stage assembly is a feature of most manufactured products
      • Modeling, analysis, and control of assembly systems is notoriously difficult
      • Few exact analytical results and the structure of the optimal policy is largely unknown
      • “ Little is known about the forms of optimal policies for multi-period models. The research to date mostly assumes particular policy types. It would be valuable to learn more about truly optimal policies. Even partial characterizations would be interesting. Also, better heuristic policy forms would be useful.”
      • --Song and Zipkin (2003)
    • Challenges
      • Demands for different items (components and sub-assemblies) are correlated
      • Production and order fulfillment depends on the availability of multiple items
      • Production leadtimes of different items can be different
      • Costs of different items can be different
    • Research Questions
      • What can we say about the structure of optimal policies?
      • What is the benefit of using optimal policies instead of common heuristics such as static base-stock policies?
      • Are there simple but effective heuristics that can serve as substitutes to optimal policies?
    • The Setting
      • Multiple items (components & intermediates) progressively assembled into a single product
      • Each item can have multiple predecessors and a single successor
      • Each item is produced (assembled) on an independent production facility
      • Demand for the end item arises from n customer classes
    • 1 2 3 4 5 6 8 9 production facility inventory location
    • Single-Stage Assembly Systems 1 2 3 4 5
    • Series Systems 1 2 3 5
    • The Demand Classes
      • Demand for the end-item emanating from class l occurs continuously one unit at a time and follows a Poisson process with rate  l
      • If demand cannot be satisfied, it is lost and incurs a lost sales cost c l
      • WLOG, c 1  c 2  …  c n
    • Item Production
      • Items are produced on independent facilities in a make-to-stock fashion
      • Item k incurs a holding cost h k ( x k ) , increasing convex in inventory level x k
      • Production times for item k are exponentially distributed with mean 1/  k
      • An item can be produced (assembled) only if at least one unit of all its predecessors items are available
      • Inventory of all items is continuously reviewed
    • A Markov Decision Process (MDP) formulation
      • System state is described by the vector X ( t )=( X 1 ( t ),…, X m ( t )) where X k ( t ) is the inventory level of item k
      • Two types of decisions are made in each state
        • Produce/not produce item k
        • Satisfy/reject an incoming order of class l
      • Objective: choose in each state the decision that minimizes (over an infinite horizon)
        • the total expected discounted cost
        • The average cost per period
    • The Optimality Equation
    • Properties of the Value Function Define:
    • Properties of the Value Function (Continued…)
    • The Optimal Production Policy
      • A base-stock production policy with state-dependent base-stock levels is optimal:
        • produce item k if x k < s k ( x - k ), x - k = ( x 1 ,…, x k -1 , x k +1 ,…, x m )
        • do not produce if x k  s k ( x - k )
      • The base-stock level s k ( x - k ) is non-increasing in x j if j  S ( k ) and is non-decreasing in x j if j  S ( k )
    • 2 3 4 5 6 7 8 9 1
    • 2 3 4 5 6 7 8 9 1 Items on the path from item 5 to item 1, S ( 5 )
    • 2 3 4 5 6 7 8 9 1 Items not on the path from item 5 to item 1
    • The Optimal Production Policy (Continued…)
      • The optimal base-stock level of item k does not decrease with the production completion of any other item
      • the closer an item in S ( k ) is to item k , the bigger the influence it has on the base-stock level of item k ,
      • for j  S ( k ) and if l  S ( j )
      • It is never optimal to interrupt the production of an item once it has been initiated
    • The Optimal Allocation Policy
      • A n allocation policy with multiple state-dependent rationing levels is optimal:
        • satisfy demand from class l if x 1  r l ( x -1 )
        • do not satisfy demand from class l if x 1 < r l ( x -1 )
      • The rationing level for class l r l ( x -1 ) is non-increasing in the inventory level x j of any item j ≠ 1
      • The rationing levels are ordered r n ( x -1 )  …  r 1 ( x -1 )=1
    • An Example with 4 Items and 3 Demand Classes 1 2 3 4 class 1 demand class 2 demand class 3 demand
    • Produce item 3 but not 2
    •  
    •  
    •  
    • x 3 s 3 ( x 1 , 3) s 3 (3, x 2 ) x 3 s 3 ( x 1 , 3)
    • Special Cases
      • Single item, single class
      • Single item, multiple classes
      • Serial system, single class
      • Two stage assembly, single class
    • Systems with Backorders (the Single Class Case)
    • The Optimal Policy
      • A base-stock policy with state-dependent base-stock levels is optimal
      • The base-stock levels retain all the properties observed in the lost sales case
    • Optimal Policy versus Heuristics
      • Base-stock policies with independent and fixed base-stock and rationing levels (IBR)
      • Dynamic linear base-stock and rationing policies (LBR)
    • Comparisons with The IBR Policy (Lost Sales)
      • For systems with lost sales, the IBR heuristic performs remarkably well (within 3% of the optimal policy in the vast majority of cases tested)
    • Comparisons with The IBR Policy (Backorders)
    • Comparisons with The IBR Policy (Backorders)
      • For systems with backorders, the IBR policy can perform poorly (an average of 13% in the cases tested and up to 40% in some cases)
      • The performance of the IBR policy is particularly poor when
        • Backorder costs are low
        • Utilization of the production facilities is high
    •  
    •  
    • Produce item 3 but not 2
    • Linear Base-Stock Production Policies (LBP)
      • Produce component k if
      • do not produce otherwise
        • The LBP policy has the same structural properties as the optimal policy
        • The LBP policy can be evaluated using simulation and the parameters s k ,  jk ,  jk obtained via a search
    • Linear Rationing Policies (LRP)
      • Satisfy demand from class l
      • do not satisfy demand otherwise
    • Summary of Work so Far
      • A formulation of the multi-stage/multi-class assembly problem
      • A characterization of the structure of the optimal policy
      • Preliminary numerical results
      • A class of simple heuristic policies
    • Ongoing and Future Work
      • A more comprehensive numerical study
      • Extensions to assembly systems with
        • demand for all items
        • multiple unit requirements
        • variable order size
        • Multiple products
        • General networks
    • Production-Inventory Systems with Imperfect Advance Demand Information and Due-Date Updates
      • Saif Benjaafar
      • Graduate Program in Industrial Engineering
      • Department of Mechanical Engineering
      • University of Minnesota
      • (Joint work with William Cooper, Jean-Philippe Gayon, Setareh Mardan , and Francis de V é ricourt )
    • Other Research
      • Production-Inventory systems with both backorders and lost sales
      • Production planning and scheduling for process industries
      • An item-customer approach to modeling, analysis and control of stochastic inventory systems
    • Comparisons with the IBR Policy (Continued…)
    •  
    •